A Biased View of Topology As a Tool for Analysis

Universidad Departamento de Murcia Matem´aticas

A biased view of topology as a tool for analysis

B. Cascales

Universidad de Murcia, Spain http://webs.um.es/beca

Trimestre Tem´aticode An´alisisFuncional, UPV. February 17th - January 18th. 2011 Valencia

Papel de Carta

Tamaño A4, reducido al 60%.

IDENTIDAD VISUAL CORPORATIVA Página 41 Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions

THE BISHOP-PHELPS-BOLLOBAS THEOREM AND ASPLUND OPERATORS

B. CASCALES

In this three lectures series we plan to present a strengthening of the Bishop-Phelps property for operators that in the literature is called the Bishop-Phelps-Bollobas´ property. Let X be a Banach space and L a locally compact Hausdorff space. We will prove that if T : X C0(L) is an Asplund operator and T (x0) T for some x0 =1, then → there is an norm attaining Asplund operator S : X C (L) and u =1with S(u ) = S = T such that → 0 0 0 u0 x0 and S T . As particular cases we obtain: (A) if T is weakly compact, then S can also be taken being weakly compact; (B) if X is Asplund (for instance, X = c0), the pair (X, C0(L)) has the Bishop-Phelps-Bollobas´ property for all L; (C) if L is scattered, the pair (X, C0(L)) has the Bishop-Phelps-Bollobas´ property for all Banach spaces X. Our idea is to present our results mostly in a self-contained way and consequently the plan will be:

Lecture 1: To recall the classical Bishop-Phelps theorem, Bollobas´ observation and their relationship with Eke- land’s variational principle; Lecture 2: To recall the notion of Asplund space, Asplund operator and establish the main ideas behind the characterization of Asplund spaces and operators via the Radon-Nikodym property and fragmentability; Lecture 3: To use the tools presented in the two previous lectures and then give a self-contained proof of the results announced in the abstract.

Key words: Bishop-Phelps, Bollobas,´ fragmentability, Asplund operator, weakly compact operator, norm attaining. AMS classiﬁcation: 46B22, 47B07

REFERENCES [1] Mar´ıa D. Acosta, Richard M. Aron, Domingo Garc´ıa, and Manuel Maestre, The Bishop-Phelps-Bollobas´ theorem for operators, J. Funct. Anal. 254 (2008), no. 11, 2780–2799. [2] Errett Bishop and R. R. Phelps, A proof that every Banach space is subreﬂexive, Bull. Amer. Math. Soc. 67 (1961), 97–98. MR MR0123174 (23 #A503) [3] Bela´ Bollobas,´ An extension to the theorem of Bishop and Phelps, Bull. London Math. Soc. 2 (1970), 181–182. MR MR0267380 (42 #2282) [4] R. M. Aron, B. Cascales and O. Kozhushkina, The Bishop-Phelps-Bollobas theorem and Asplund operators, to appear PAMS 2010-2011. [5] Jerry Johnson and John Wolfe, Norm attaining operators, Studia Math. 65 (1979), no. 1, 7–19. [6] I. Namioka. Fragmentability in Banach spaces: Interaction of topologies, RACSAM 104 (2), 2010, 283-308. [7] C. Stegall, The Radon-Nikodym´ property in conjugate Banach spaces. II, Trans. Amer. Math. Soc. 264 (1981), no. 2, 507–519.

B. Cascales Departamento de Matematicas´ Universidad de Murcia 30100 Espinardo. Murcia. Spain [email protected]

1 Universidad Facultad de de Murcia Matem´aticas

A biased view of topology as a tool for analysis Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions B. Cascales

When presenting these lectures we are strongly motivated by the fact that general topology and functional analysis continuously beneﬁt from cross-fertilization between them. Our starting point for these two lectures, intended for students, are the two exercises below.

Exercise 1 A compact Hausdorﬀ topological space K is metrizable if, and only, if (C(K), ) is separable. · ∞ Exercise 2 From Engelking’s book [3]:

Both exercises are connected. From Exercise 1 we will motivate some classical results about weak compactness in Banach spaces. Exercise 2 can be easily rephrased as follows: a compact Hausdorff topological space K is metrizable if, and only if, (K K) ∆ = × \ n N Fn with each Fn a closed subset of K K. From here we will move to some other ∈ × N more intriguing cases. To name one, if (K K) ∆ = Aα : α N where each Aα is × \ { ∈ } compact and Aα Aβ whenever α β, we shall prove that the latter assumption also ⊂ ≤ N implies metrizability when either Aα : α N is a fundamental family of compact { ∈ } subsets for (K K) ∆ or when MA(ω ) is assumed. The success when proving these × \ 1 results relies upon the generation of usco maps. We provide applications (old and new) of the results and techniques presented here to functional analysis: metrizability of compact subsets in inductive limits, Lindelöfproperty of WCG Banach spaces and classification of compact topological spaces, separability of Fréchet-Montel spaces, Lindelöf-Σ character of spaces Cp(X), etc. For the students is a good objective to learn all the details of how to solve both exercises. Furthermore, the lectures will stress on how these simple but tricky ideas have motivated recent Ph. D. dissertations as a well as some new results and applications published elsewhere. Referencias

[1] B. Cascales and J. Orihuela, On compactness in locally convex spaces,Math.Z.195 (1987), no. 3, 365–381. [2] B. Cascales, J. Orihuela and V. Tkachuk, Domination by second countable spaces,Toappear in Top. and its Appl., 2011. [3] R. Engelking. General topology. PWN—Polish Scientiﬁc Publishers, Warsaw, 1977. [Mathe- matical Monographs, Vol. 60]. [4] J. K¸akol, W. Kubi´sand M. L´opez-Pellicer, Descriptive topology in selected topics of functional analysis, 628 pages. Preprint, 2011.

B. Cascales. Departamento de Matem´aticas. Campus de Espinardo. 30100 Espinardo Murcia Tfno. 868884174. e-mail: [email protected]:http://webs.um.es/beca

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1 The proto-ideas

2 Starting Point

3 Domination by Polish Spaces: applications to FA

4 Domination by Second Countable Spaces

5 Further extensions and open questions Ω C open set; H (Ω) space of holomorphic functions with ⊂ the topology of uniform convergence on compact sets; n Ω R open set; D0(Ω) space of distributions; ⊂ limEn inductive limit of a sequence of Fr´echetspaces. −→

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Notation

L,M,...X ,Y ,... topological spaces; E,F Banach or sometimes lcs; K compact Hausdorﬀ space; 2X subsets; K (X ) family of compact sets;

C(X ) continuous functions; Cp(X ) continuous functions endowed with the pointwise convergence topology τp; Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Notation

L,M,...X ,Y ,... topological spaces; E,F Banach or sometimes lcs; K compact Hausdorﬀ space; 2X subsets; K (X ) family of compact sets;

C(X ) continuous functions; Cp(X ) continuous functions endowed with the pointwise convergence topology τp; Ω C open set; H (Ω) space of holomorphic functions with ⊂ the topology of uniform convergence on compact sets; n Ω R open set; D0(Ω) space of distributions; ⊂ limEn inductive limit of a sequence of Fr´echetspaces. −→ Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions First of Two inspiring papers

Valdivia, J. London Math. Soc.QUASI-LB-SPACE 1987 S 151 Let F be a Frechet space and let {Un: n = 1,2,:..} be a fundamental system of neighbourhoods of the origin in F which we can take to be closed and absolutely N convex. If a = (an) is any element of N , we set

Aa = f]{anUn:n=\,2,...}. QUASI-LB-SPACES The set Aa is a Banach disc and we have the following properties. N N (1) The family {^a: ae N } covers F. (2) If a,0e N and a ^ /?, then Aa a Ap. Let us now suppose that FMANUE is an LB-spaceL VALDIVI. ThereA is an increasing sequence Bn of N Banach discs covering F. For every a = (an)eN , we set Aa = Bai; then the family N {^a:aeM } has properties (1) and (2) also. We shall see later that properties Introduction(1) and (2) are important in order to obtain some results on the closed graph theorem. This is the reason for introducing the following definitionsGrothendiec. A quasi-LB-representationk conjectured in [4] tha int th ae topologicalclosed grap vectorh theore spacem hold F sis whe a familyn the {Adomai: aen NspacN} eof isBanach ultrabornologica discs satisfyingl and thethe following range spac conditions:e belongs to a class of locally convea x spaces containing the Banach spaces which are stable under the following operations: countable topological products, countable topological direct sums, separated quotients and closed subspaces. In particular, if Q is a non-void open subset 2. //"a,/?e NN anda^P then A c A . of ^-dimensional euclidean space, thea n 3>(Q)fi and <2)'{£l) belong to this class. In [8], SlowikowskWe shall sayi definethat ad space a clas admittings of range aspace quasi-LB-representations that contains $)(Q) isan ad 2)'{QL)quasi-LB-space. and verified the closed graph theorem when the domain space is a Banach space. Rafkow [7] gives M.a Valdivia, solutioPROPOSITIOn to Grothendieck'NQuasi-LB-spaces 1. Let F sbe conjecture a quasi-LB-space., usin,g J. to somLet London eG exten be at spacethe Math. idea suchs of thatSlowikowski Soc. there is (2) a. 35 Dcontinuouse Wilde givelinears i nmapping [2] a positiv from e Fanswe onto rG. to Then the GrothendiecG is a quasi-LB-space.k conjecture for webbed (1987),spaces. no.In this 1, articl 149–168.e we introduc MRe quasi-LB-spaces 88b:46012, which form a subclass, with a simple definition, of thNe class of Slowikowski; these spaces also answer the Proof. Let {Aa: ae N } be a quasi-LB-representation of F. It is easy to show that Grothendieck conjecture. We proved in [12] that the Slowikowski spaces coincide with those of Rafkov, and that the strict webbed spaces are Slowikowski also. A iquasi-LB-spacs a quasi-LB-representatioe is a strict webben of G.d space, and a locally complete webbed space is a quasi-LB-space and thus a strict webbed space. This property, that answers an old question posed by De Wilde, allows us to simplify the theory of webbed spaces PROPOSITION 2. The countable product of quasi-LB-spaces is a quasi-LB-space. developed by De Wilde, looking at larger and different aspects than he does. We have introduced the quasi-LB-spaces because of their simple definition and good properties, and Proof.becaus e Fother yeac shoh jw i nu pa thgivee underlyinn countablg estructur set /, wee odenotf the eclose by ^d agrap spach etheore admittinm ing a forquasi-LB-representatiom that we believe ton be very clear, and that is related to the classical Banach theorem [1]. This structure is shown clearly in the lifting and localization theorems thaWet writwe epresent in this article as in the method of answering De Wilde's question.

F=U{Fj:jeJ}. 2. Terminology and notation N Let be a one to one mapping from N onto JxN. For o^ = (a^ n)e M ,jeJ, we put The vector spaces we shall use here are defineN d over the field K of real or complex df(n) — bn, n= 1,2,..., {//(ocj-.jeJ}) = (bn)eN . Then y/ is a one to one mapping numbersfrom (NN. )WJ eont supposo N*e1 .tha If ta ATiis san endowey elemend witt ohf th^ e ordinarand {(XfjeJ)y topology = y/-\

Received 10 February 1986. 1980 Mathematics Subject Classification 46A30. J. London Math. Soc. (2) 35 (1987) 149-168 Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Second of Two inspiring papers

Talagrand, Ann. of Math. 1979

M. Talagrand, Espaces de Banach faiblement K -analytiques, Ann. of Math. (2) 110 (1979), no. 3, 407–438. MR 81a:46021 2 If αn α in NN then there is β NN such that → ∈

αn,α β ≤ (here stands for the natural order for the coordinates) ≤

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Simple facts to keep in mind

1 NN endowed with the product of discrete topology on N is separable and metrizable with a complete metric (i.e. NN is a Polish space). Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Simple facts to keep in mind

1 NN endowed with the product of discrete topology on N is separable and metrizable with a complete metric (i.e. NN is a Polish space). 2 If αn α in NN then there is β NN such that → ∈

αn,α β ≤ (here stands for the natural order for the coordinates) ≤ As good as the need/use of them for applications.

2 ¿How good are the results?

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Kind of results to be presented

1 Structures related to descriptive set theory that apply often to Functional Analysis; As good as the need/use of them for applications.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Kind of results to be presented

1 Structures related to descriptive set theory that apply often to Functional Analysis;

2 ¿How good are the results? Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Kind of results to be presented

1 Structures related to descriptive set theory that apply often to Functional Analysis;

2 ¿How good are the results? As good as the need/use of them for applications. Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions A few words about descriptive set theory

Descriptive set theory From Wikipedia, the free encyclopedia In mathematical logic, descriptive set theory is the study of certain classes of ”well-behaved” subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic. Proto Idea Starting Point Domination by Polish&Appl. DominationDescriptive byset theory second - Wikipedia, countable the freeExtensions&Open encyclopedia questions http://en.wikipedia.org/wiki/Descriptive_set_theory

A few words about descriptive setDescriptive theory set theory From Wikipedia, the free encyclopedia

In mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic. Descriptive set theory From Wikipedia, the free encyclopedia Contents In mathematical logic, descriptive set theory 1 Polish spaces 1.1 Universality properties is the study of certain classes of 2 Borel sets ”well-behaved” subsets of the real line and 2.1 Borel hierarchy other Polish spaces. As well as being one of 2.2 Regularity properties of Borel sets the primary areas of research in set theory, 3 Analytic and coanalytic sets 4 Projective sets and Wadge degrees it has applications to other areas of 5 Borel equivalence relations mathematics such as functional analysis, 6 Effective descriptive set theory 7 See also ergodic theory, the study of operator 8 References algebras and group actions, and 9 External links mathematical logic. Polish spaces

Descriptive set theory begins with the study of Polish spaces and their Borel sets.

A Polish space is a second countable topological space that is metrizable with a complete metric. Equivalently, it is a complete separable metric space whose metric has been "forgotten". Examples include the real line , the Baire space , the Cantor space , and the Hilbert cube .

Universality properties

The class of Polish spaces has several universality properties, which show that there is no loss of generality in considering Polish spaces of certain restricted forms.

Every Polish space is homeomorphic to a G! subspace of the Hilbert cube, and every G! subspace of the Hilbert cube is Polish. Every Polish space is obtained as a continuous image of Baire space; in fact every Polish space is the image of a continuous bijection deﬁned on a closed subset of Baire space. Similarly, every compact Polish space is a continuous image of Cantor space.

Because of these universality properties, and because the Baire space has the convenient property that it is homeomorphic to , many results in descriptive set theory are proved in the context of Baire space alone.

1 of 4 2/17/11 3:04 AM I93I-] SHORTER NOTICES 507

plete mathematical references are given in an appendix. The same is true of a number of other purely mathematical topics, such as complex integration, orthogonal functions, curvilinear coordinates, etc. A whole chapter is devoted to the study of hydrogen atoms by the wave mechanics method with sufficient concrete details and applications to make clear to the student the utility of the method. The matrix mechanics is then introduced and developed as an independent theory, with detailed study of applications. This is followed by a discussion of the connection between the wave and matrix mechanics, and the method of constructing the quantum matrices from the solutions of the wave equation. In the chapter on the general theory of quantum dynamics, there is a thorough treatment of Heisenberg's indétermination principle, and the transformation theory of Jordan and Dirac. The frequent introduction of concrete illustrations greatly enhances the value of this chapter. The book is concluded with chapters on the treatment of non-hydrogenic atoms and molecules by the new mechanics, spectral intensities and the diffrac- tion of electrons and atoms by crystals. The style of the book is in general clear and concise. The typography is Proto IdeaexcellenStartingt an Pointd theDomination text isby wel Polish&Appl.l illustrateDominationd by a bylarg seconde numbe countabler oExtensions&Openf well-made questionsdia- grams. On the whole it is a work which may be heartily recommended to all those interested in the problems of atomic structure. The origin of descriptive set theory R. B. LINDSAY

Leçons sur les Ensembles Analytiques et leurs Applications. By Nicolas Lusin, With a preface by Henri Lebesgue and a note by Waclaw Sierpinski. Paris, Gauthier-Villars, 1930. xvi+328 pages. This volume in the Borel series contains a systematic survey of the present knowledge of analytic sets, a knowledge which is chiefly due to the researches of the Russian mathematician who is the author of this book. In fact the only results which are not due to Lusin or his pupils come from members of the Polish school of Sierpinski and Mazurkiewicz. The analytic sets of Lusin, which are a generalization of Borel sets, have been briefly mentioned previously in several books (Hausdorff's Mengenlehre, for instance), but this is the first book devoted entirely to their study. Lebesgue in his preface humorously points out that the origin of the problems considered by Lusin lies in an error made by Lebesgue himself in his 1905 memoir on functions representable analytically. Lebesgue stated there that the projection of a Borel set is always a Borel set. Lusin and his colleague Souslin constructed an example showing that this statement was false, thus discovering a new domain of point sets, a domain which includes as a proper part the domain of Borel sets. Lebesgue expresses his joy that he was inspired to commit such a fruitful error. Of the five chapters of approximately equal length into which the book is divided, the first two are devoted to Borel sets. Here and throughout the book, Lusin considers as his fundamental domain the set of irrational points of a linear space. By excluding the rational points, certain simplifications in statements and proofs of theorems are obtained. After mentioning several different methods of defining Borel sets and showing their logical equivalence, a study is Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions A few definitions from descriptive set theory Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions A few definitions from descriptive set theory Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions A few definitions from descriptive set theory

D. L. Cohn, Measure theory, Birkh¨auser,Boston, Mass., 1980. MR 81k:28001 Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions

Starting point 1 Let (E, ) a Banach space and BE the unit dual ball. Then, k k ∗ (BE ,w ) metrizable, if and only if, (E, ) is separable. ∗ ∗ k k 2 Let (E, ) be a Banach space. Then, (BE ,w)) is metrizable k k if, and only if, (E ) is separable. ∗k k 3 (Smulian,ˇ 1940) Let E be a Banach space. The w-compact subsets of E are w-sequentially compact, i.e., if H E ⊂ w-compact, then each sequence (xn)n en H has a subsequence that w-converges to a point in H.

Results in FA in the “same family”:

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions A characterization of metrizability

Exercise A compact Hausdorﬀ topological space K is metrizable if, and only, if (C(K), ∞) is separable. k · k 1 Let (E, ) a Banach space and BE the unit dual ball. Then, k k ∗ (BE ,w ) metrizable, if and only if, (E, ) is separable. ∗ ∗ k k 2 Let (E, ) be a Banach space. Then, (BE ,w)) is metrizable k k if, and only if, (E ) is separable. ∗k k 3 (Smulian,ˇ 1940) Let E be a Banach space. The w-compact subsets of E are w-sequentially compact, i.e., if H E ⊂ w-compact, then each sequence (xn)n en H has a subsequence that w-converges to a point in H.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions A characterization of metrizability

Exercise A compact Hausdorﬀ topological space K is metrizable if, and only, if (C(K), ∞) is separable. k · k Results in FA in the “same family”: 2 Let (E, ) be a Banach space. Then, (BE ,w)) is metrizable k k if, and only if, (E ) is separable. ∗k k 3 (Smulian,ˇ 1940) Let E be a Banach space. The w-compact subsets of E are w-sequentially compact, i.e., if H E ⊂ w-compact, then each sequence (xn)n en H has a subsequence that w-converges to a point in H.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions A characterization of metrizability

Exercise A compact Hausdorﬀ topological space K is metrizable if, and only, if (C(K), ∞) is separable. k · k Results in FA in the “same family”:

1 Let (E, ) a Banach space and BE the unit dual ball. Then, k k ∗ (BE ,w ) metrizable, if and only if, (E, ) is separable. ∗ ∗ k k 3 (Smulian,ˇ 1940) Let E be a Banach space. The w-compact subsets of E are w-sequentially compact, i.e., if H E ⊂ w-compact, then each sequence (xn)n en H has a subsequence that w-converges to a point in H.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions A characterization of metrizability

Exercise A compact Hausdorﬀ topological space K is metrizable if, and only, if (C(K), ∞) is separable. k · k Results in FA in the “same family”:

1 Let (E, ) a Banach space and BE the unit dual ball. Then, k k ∗ (BE ,w ) metrizable, if and only if, (E, ) is separable. ∗ ∗ k k 2 Let (E, ) be a Banach space. Then, (BE ,w)) is metrizable k k if, and only if, (E ) is separable. ∗k k Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions A characterization of metrizability

Exercise A compact Hausdorﬀ topological space K is metrizable if, and only, if (C(K), ∞) is separable. k · k Results in FA in the “same family”:

1 Let (E, ) a Banach space and BE the unit dual ball. Then, k k ∗ (BE ,w ) metrizable, if and only if, (E, ) is separable. ∗ ∗ k k 2 Let (E, ) be a Banach space. Then, (BE ,w)) is metrizable k k if, and only if, (E ) is separable. ∗k k 3 (Smulian,ˇ 1940) Let E be a Banach space. The w-compact subsets of E are w-sequentially compact, i.e., if H E ⊂ w-compact, then each sequence (xn)n en H has a subsequence that w-converges to a point in H. Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Another characterization of metrizability

Exercise. . . from Engelking’s book Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Another characterization of metrizability

Exercise. . . from Engelking’s book 2 (C(K), ∞) is separable; k · k 3 ∆ is a Gδ ; 4 ∆ = nGn with Gn open and Gn n a basis of neighb. of ∆; ∩ { } 5 (K K) ∆ = nFn, with Fn an increasing fundamental × \ ∪ { } family of compact sets in (K K) ∆; × \ N 6 (K K) ∆ = Aα : α N with each Aα a fundamental × \ { ∈ } { } family of compact sets such that Aα A whenever α β; S ⊂ β ≤ 7 (K K) ∆ is Lindel¨of. × \

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions We are indeed a bit more greedy

The goal For a compact space K TFAE: 1 K is metrizable; 3 ∆ is a Gδ ; 4 ∆ = nGn with Gn open and Gn n a basis of neighb. of ∆; ∩ { } 5 (K K) ∆ = nFn, with Fn an increasing fundamental × \ ∪ { } family of compact sets in (K K) ∆; × \ N 6 (K K) ∆ = Aα : α N with each Aα a fundamental × \ { ∈ } { } family of compact sets such that Aα A whenever α β; S ⊂ β ≤ 7 (K K) ∆ is Lindel¨of. × \

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions We are indeed a bit more greedy

The goal For a compact space K TFAE: 1 K is metrizable;

2 (C(K), ∞) is separable; k · k 4 ∆ = nGn with Gn open and Gn n a basis of neighb. of ∆; ∩ { } 5 (K K) ∆ = nFn, with Fn an increasing fundamental × \ ∪ { } family of compact sets in (K K) ∆; × \ N 6 (K K) ∆ = Aα : α N with each Aα a fundamental × \ { ∈ } { } family of compact sets such that Aα A whenever α β; S ⊂ β ≤ 7 (K K) ∆ is Lindel¨of. × \

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions We are indeed a bit more greedy

The goal For a compact space K TFAE: 1 K is metrizable;

2 (C(K), ∞) is separable; k · k 3 ∆ is a Gδ ; 5 (K K) ∆ = nFn, with Fn an increasing fundamental × \ ∪ { } family of compact sets in (K K) ∆; × \ N 6 (K K) ∆ = Aα : α N with each Aα a fundamental × \ { ∈ } { } family of compact sets such that Aα A whenever α β; S ⊂ β ≤ 7 (K K) ∆ is Lindel¨of. × \

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions We are indeed a bit more greedy

The goal For a compact space K TFAE: 1 K is metrizable;

2 (C(K), ∞) is separable; k · k 3 ∆ is a Gδ ; 4 ∆ = nGn with Gn open and Gn n a basis of neighb. of ∆; ∩ { } N 6 (K K) ∆ = Aα : α N with each Aα a fundamental × \ { ∈ } { } family of compact sets such that Aα A whenever α β; S ⊂ β ≤ 7 (K K) ∆ is Lindel¨of. × \

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions We are indeed a bit more greedy

The goal For a compact space K TFAE: 1 K is metrizable;

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions We are indeed a bit more greedy

The goal For a compact space K TFAE: 1 K is metrizable;

2 (C(K), ∞) is separable; k · k 3 ∆ is a Gδ ; 4 ∆ = nGn with Gn open and Gn n a basis of neighb. of ∆; ∩ { } 5 (K K) ∆ = nFn, with Fn an increasing fundamental × \ ∪ { } family of compact sets in (K K) ∆; × \ N 6 (K K) ∆ = Aα : α N with each Aα a fundamental × \ { ∈ } { } family of compact sets such that Aα A whenever α β; S ⊂ β ≤ Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions We are indeed a bit more greedy

The goal For a compact space K TFAE: 1 K is metrizable;

2 (C(K), ∞) is separable; k · k 3 ∆ is a Gδ ; 4 ∆ = nGn with Gn open and Gn n a basis of neighb. of ∆; ∩ { } 5 (K K) ∆ = nFn, with Fn an increasing fundamental × \ ∪ { } family of compact sets in (K K) ∆; × \ N 6 (K K) ∆ = Aα : α N with each Aα a fundamental × \ { ∈ } { } family of compact sets such that Aα A whenever α β; S ⊂ β ≤ 7 (K K) ∆ is Lindel¨of. × \ Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions The proof. . .

The goal

For a compact space K TFAE: 1 K is metrizable;

2 (C(K), ∞) is separable; k · k 3 ∆ is a Gδ ; 4 ∆ = nGn with Gn open and Gn n a basis of neighb. of ∆; ∩ { } 5 (K K) ∆ = nFn, with Fn an increasiang fundamental family of compact sets in (K K) ∆; × \ ∪ { } × \ 6 (K K) ∆ = Aα : α NN with each Aα a fundamental family of compact sets such that Aα A × \ { ∈ } { } ⊂ β whenever α β; ≤ S 7 (K K) ∆ is Lindelöf. × \ For Kα Dβ = , we denote by yαβ the element of X such that Kα Dβ = y . If ∩ ∅ ∩ { αβ} N N T := (α, β) N N : = Kα Dβ = yαβ , ∈ × ∅ ∩ { } the map f : T X defined by f((α, β)) = y is surjection. → αβ Let (α(p), β(p)) be a sequence in T that converges to (α, β) in NN NN, p × and let (α(p), β(p))p(m) be a subsequence. By the K-analyticity y has an adherent point y K . Since α(p),β(p) p(m) ∈ α N β(p) converges to β =(bn)n N , the sequence yα(p),β(p) is eventually ∈ p(m) in each B (z ,n 1), hence its adherent point y belongs to B (z ,n 1). d bn − d bn − This shows that Proto Idea Starting Point Domination by Polish&Appl.y K DDomination= y by. second countable Extensions&Open questions ∈ α ∩ β { αβ} We proved that (α, β) T , i.e., T is a closed subset of NN NN and, therefore, (K K) ∆ is Lindelöf∈ ∆ is a G × × T is\ a Polish space. Moreover,⇒ we proved thatδ yαβ is an adherent point of each subsequence of yα(p),β(p) p. This implies that yα(p),β(p) converges to yαβ, i.e., f (α(p), β(p)) converges to f (α, β). Hence f is a continuous mapping from the Polish space T onto (Y,τ), and this proves that (Y,τ) is analytic. In order to prove the second part of the proposition assume that ∆ = It works(x, x): evenx X for isX theHausdorff diagonal of regular the analytic space. space X X. Clearly ∆ and ({X X) ∆∈ are} analytic and, therefore, they are Lindelöf.× × \ If x = y, there exist two closed neighbourhoods Fx and Fy of x and y, respective, such that F F (X X) ∆. x × y ⊂ × \ The Lindelöf property enables to determine a sequence (xn,yn)n such that

X X ∆ = F F . × \ xn × yn n Therefore ∆ is a G -subset of X X since ∆ = G , where δ × n n G =(X X) (F F). n × \ xn × yn

For each (x, x) ∆ and n N there exists an open set Ux,n in X such that ∈ ∈ (x, x) U U G . ∈ x,n × x,n ⊂ n 173 Y 1 Y is K-analytic if there is ψ : NN 2 that is upper semi-continuous compact-valued and such that Y =→ ( ); α NN ψ α ∈ Y 2 Y is countably K-determined if there is Σ NN and ψ :Σ 2 that is S ⊂ → upper semi-continuous compact-valued and such that Y = α Σ ψ(α). ∈ S

Σ any second countable space M (Lindel¨ofΣ) ⇔ NN any Polish space P ⇔

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Deﬁnitions

Upper semi-continuous set-valued map (multi-function) ψ X - 2Y

ψ(x) x j

ψ(x0) : U V Σ any second countable space M (Lindel¨ofΣ) ⇔ Y 2 Y is countably K-determined if there is Σ NN and ψ :Σ 2 that is N any Polish space⊂ P → upper semi-continuousN compact-valued and such that Y = α Σ ψ(α). ⇔ ∈ S

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Deﬁnitions

Upper semi-continuous set-valued map (multi-function) ψ X - 2Y

ψ(x) x j

ψ(x0) : U V

Y 1 Y is K-analytic if there is ψ : NN 2 that is upper semi-continuous compact-valued and such that Y =→ ( ); α NN ψ α ∈ S Σ any second countable space M (Lindel¨ofΣ) ⇔ NN any Polish space P ⇔

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Deﬁnitions

Upper semi-continuous set-valued map (multi-function) ψ X - 2Y

ψ(x) x j

ψ(x0) : U V

Y 1 Y is K-analytic if there is ψ : NN 2 that is upper semi-continuous compact-valued and such that Y =→ ( ); α NN ψ α ∈ Y 2 Y is countably K-determined if there is Σ NN and ψ :Σ 2 that is S ⊂ → upper semi-continuous compact-valued and such that Y = α Σ ψ(α). ∈ S Σ any second countable space M (Lindel¨ofΣ) ⇔

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Deﬁnitions

Upper semi-continuous set-valued map (multi-function) ψ X - 2Y

ψ(x) x j

ψ(x0) : U V

Y 1 Y is K-analytic if there is ψ : NN 2 that is upper semi-continuous compact-valued and such that Y =→ ( ); α NN ψ α ∈ Y 2 Y is countably K-determined if there is Σ NN and ψ :Σ 2 that is N any PolishS space⊂ P → upper semi-continuousN compact-valued and such that Y = α Σ ψ(α). ⇔ ∈ S NN any Polish space P ⇔

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Deﬁnitions

Upper semi-continuous set-valued map (multi-function) ψ X - 2Y

ψ(x) x j

ψ(x0) : U V

1 N Y Y is ΣK-analyticany ifsecond there is countableψ : N 2 spacethatM is upper(LindelöfΣ) semi-continuous compact-valued and such that Y =→ ( ); ⇔ α NN ψ α ∈ Y 2 Y is countably K-determined if there is Σ NN and ψ :Σ 2 that is S ⊂ → upper semi-continuous compact-valued and such that Y = α Σ ψ(α). ∈ S 2 if ψ : X 2Y that is upper semi-continuous compact-valued, then L X is→ Lindelöf ψ(L) is Lindelöf; ⊂ ⇒ 3 K-analytic countably K-determined Lindelöf; ⇒ ⇒ 4 countably K-determined + metrizable separable; X ⇒ 5 if X is K-analytic (ψ : NN 2 ) and Aα := ψ( β : β α ) then: → { ≤ } (A) each Aα is compact; (B) Aα Aβ whenever α β; ⊂ N ≤ (C) X = Aα : α N . { ∈ } 6 ditto, if X Sis countably K-determined, there is a second countable space M and a family AK : K K (M) such that: { ∈ } (A) each AK is compact; (B) AK AF whenever K F ; ⊂ ⊂ (C) X = AK : K K (M) . { ∈ } S

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Easy known facts

1 If ψ : X 2Y that is upper semi-continuous compact-valued, then K X is→ compact ψ(K) is compact; ⊂ ⇒ 3 K-analytic countably K-determined Lindel¨of; ⇒ ⇒ 4 countably K-determined + metrizable separable; X ⇒ 5 if X is K-analytic (ψ : NN 2 ) and Aα := ψ( β : β α ) then: → { ≤ } (A) each Aα is compact; (B) Aα Aβ whenever α β; ⊂ N ≤ (C) X = Aα : α N . { ∈ } 6 ditto, if X Sis countably K-determined, there is a second countable space M and a family AK : K K (M) such that: { ∈ } (A) each AK is compact; (B) AK AF whenever K F ; ⊂ ⊂ (C) X = AK : K K (M) . { ∈ } S

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Easy known facts

1 If ψ : X 2Y that is upper semi-continuous compact-valued, then K X is→ compact ψ(K) is compact; ⊂ ⇒ 2 if ψ : X 2Y that is upper semi-continuous compact-valued, then L X is→ Lindel¨of ψ(L) is Lindel¨of; ⊂ ⇒ 4 countably K-determined + metrizable separable; X ⇒ 5 if X is K-analytic (ψ : NN 2 ) and Aα := ψ( β : β α ) then: → { ≤ } (A) each Aα is compact; (B) Aα Aβ whenever α β; ⊂ N ≤ (C) X = Aα : α N . { ∈ } 6 ditto, if X Sis countably K-determined, there is a second countable space M and a family AK : K K (M) such that: { ∈ } (A) each AK is compact; (B) AK AF whenever K F ; ⊂ ⊂ (C) X = AK : K K (M) . { ∈ } S

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Easy known facts

1 If ψ : X 2Y that is upper semi-continuous compact-valued, then K X is→ compact ψ(K) is compact; ⊂ ⇒ 2 if ψ : X 2Y that is upper semi-continuous compact-valued, then L X is→ Lindelöf ψ(L) is Lindelöf; ⊂ ⇒ 3 K-analytic countably K-determined Lindelöf; ⇒ ⇒ X 5 if X is K-analytic (ψ : NN 2 ) and Aα := ψ( β : β α ) then: → { ≤ } (A) each Aα is compact; (B) Aα Aβ whenever α β; ⊂ N ≤ (C) X = Aα : α N . { ∈ } 6 ditto, if X Sis countably K-determined, there is a second countable space M and a family AK : K K (M) such that: { ∈ } (A) each AK is compact; (B) AK AF whenever K F ; ⊂ ⊂ (C) X = AK : K K (M) . { ∈ } S

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Easy known facts

1 If ψ : X 2Y that is upper semi-continuous compact-valued, then K X is→ compact ψ(K) is compact; ⊂ ⇒ 2 if ψ : X 2Y that is upper semi-continuous compact-valued, then L X is→ Lindelöf ψ(L) is Lindelöf; ⊂ ⇒ 3 K-analytic countably K-determined Lindelöf; ⇒ ⇒ 4 countably K-determined + metrizable separable; X ⇒ 5 if X is K-analytic (ψ : NN 2 ) and Aα := ψ( β : β α ) then: → { ≤ } (A) each Aα is compact; (B) Aα Aβ whenever α β; ⊂ N ≤ (C) X = Aα : α N . { ∈ } S Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Easy known facts

1 If ψ : X 2Y that is upper semi-continuous compact-valued, then K X is→ compact ψ(K) is compact; ⊂ ⇒ 2 if ψ : X 2Y that is upper semi-continuous compact-valued, then L X is→ Lindelöf ψ(L) is Lindelöf; ⊂ ⇒ 3 K-analytic countably K-determined Lindelöf; ⇒ ⇒ 4 countably K-determined + metrizable separable; X ⇒ 5 if X is K-analytic (ψ : NN 2 ) and Aα := ψ( β : β α ) then: → { ≤ } (A) each Aα is compact; (B) Aα Aβ whenever α β; ⊂ N ≤ (C) X = Aα : α N . { ∈ } 6 ditto, if X Sis countably K-determined, there is a second countable space M and a family AK : K K (M) such that: { ∈ } (A) each AK is compact; (B) AK AF whenever K F ; ⊂ ⊂ (C) X = AK : K K (M) . { ∈ } S Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions To keep in mind

Proposition Let X be a metric space and ψ : X 2Y multi-valued . TFAE: → 1 ψ is usco;

2 ψ is compact valued + For every sequence xn x in X if → yn ψ(xn), n N then (yn)n has a cluster point y ψ(x). ∈ ∈ ∈ Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions

Domination by Polish Spaces Proposition, Orihuela-Tkachuk-C, 2011 For a topological space X the TFAE: 1 X is dominated by a Polish space; N 2 There is a family Aα : α N of subsets of X with: { ∈ } (A) each Aα is compact; (B) Aα Aβ whenever α β; ⊂ N ≤ (C) X = Aα : α N . { ∈ } S

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Domination by Polish Spaces

Deﬁnition A topological space X is dominated by a Polish space, if there is a Polish space P and a family AK : K K (P) X such that: { ∈ } ⊂ (A) each AK is compact;

(B) AK AF whenever K F ; ⊂ ⊂ (C) X = AK : K K (P) . { ∈ } S Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Domination by Polish Spaces

Deﬁnition A topological space X is dominated by a Polish space, if there is a Polish space P and a family AK : K K (P) X such that: { ∈ } ⊂ (A) each AK is compact;

(B) AK AF whenever K F ; ⊂ ⊂ (C) X = AK : K K (P) . { ∈ } S Proposition, Orihuela-Tkachuk-C, 2011 For a topological space X the TFAE: 1 X is dominated by a Polish space; N 2 There is a family Aα : α N of subsets of X with: { ∈ } (A) each Aα is compact; (B) Aα Aβ whenever α β; ⊂ N ≤ (C) X = Aα : α N . { ∈ } S Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Two nice previous cases

Talagrand, Ann. of Math. 1979 Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Two nice previous cases

Aa = f]{anUn:n=\,2,...}. QUASI-LB-SPACES The set Aa is a Banach disc and we have the following properties. N N (1) The family {^a: ae N } covers F. (2) If a,0e N and a ^ /?, then Aa a Ap. Let us now suppose that FMANUE is an LB-spaceL VALDIVI. ThereA is an increasing sequence Bn of N Banach discs covering F. For every a = (an)eN , we set Aa = Bai; then the family N {^a:aeM } has properties (1) and (2) also. We shall see later that properties Introduction(1) and (2) are important in order to obtain some results on the closed graph theorem. This is the reason for introducing the following definitionsGrothendiec. A quasi-LB-representationk conjectured in [4] tha int th ae topologicalclosed grap vectorh theore spacem hold F sis whe a familyn the {Adomai: aen NspacN} eof isBanach ultrabornologica discs satisfyingl and thethe following range spac conditions:e belongs to a class of locally convea x spaces containing the Banach spaces which are stable under the following operations: countable topological products, countable topological direct sums, separated quotients and closed subspaces. In particular, if Q is a non-void open subset 2. //"a,/?e NN anda^P then A c A . of ^-dimensional euclidean space, thea n 3>(Q)fi and <2)'{£l) belong to this class. In [8], SlowikowskWe shall sayi definethat ad space a clas admittings of range aspace quasi-LB-representations that contains $)(Q) isan ad 2)'{QL)quasi-LB-space. and verified the closed graph theorem when the domain space is a Banach space. Rafkow [7] gives a solutioPROPOSITIOn to Grothendieck'N 1. Let F sbe conjecture a quasi-LB-space., using to somLet eG exten be at spacethe idea suchs of thatSlowikowski there is a. Dcontinuouse Wilde givelinears i nmapping [2] a positiv from e Fanswe onto rG. to Then the GrothendiecG is a quasi-LB-space.k conjecture for webbed spaces. In this article we introduce quasi-LB-spaces, which form a subclass, with a simple definition, of thNe class of Slowikowski; these spaces also answer the Proof. Let {Aa: ae N } be a quasi-LB-representation of F. It is easy to show that Grothendieck conjecture. We proved in [12] that the Slowikowski spaces coincide with those of Rafkov, and that the strict webbed spaces are Slowikowski also. A iquasi-LB-spacs a quasi-LB-representatioe is a strict webben of G.d space, and a locally complete webbed space is a quasi-LB-space and thus a strict webbed space. This property, that answers an old question posed by De Wilde, allows us to simplify the theory of webbed spaces PROPOSITION 2. The countable product of quasi-LB-spaces is a quasi-LB-space. developed by De Wilde, looking at larger and different aspects than he does. We have introduced the quasi-LB-spaces because of their simple definition and good properties, and Proof.becaus e Fother yeac shoh jw i nu pa thgivee underlyinn countablg estructur set /, wee odenotf the eclose by ^d agrap spach etheore admittinm ing a forquasi-LB-representatiom that we believe ton be very clear, and that is related to the classical Banach theorem [1]. This structure is shown clearly in the lifting and localization theorems thaWet writwe epresent in this article as in the method of answering De Wilde's question.

Received 10 February 1986. 1980 Mathematics Subject Classification 46A30. J. London Math. Soc. (2) 35 (1987) 149-168 Given α = (nk ) NN and m N, deﬁne ∈ ∈ α m := (n1,n2,...,nm). | Proposition, B. C., 1987 X Given X and Aα : α NN as above, if we deﬁne ϕ : NN 2 given by { ∈ } ∞ → ϕ(α) := A : β k = α k { β | | } then: k\=1 [ each ϕ(α) is countably compact (even more, all cluster points of any sequence in ϕ(α) remains in ϕ(α)). if ϕ(α) is compact then α ϕ(α) gives K-analytic structure to X . →

X has K-analytic structure if countably compact subsets=compact subsets.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Domination by Polish implies (many times) K-analyticity

Let X be a topological space Aα : α NN of subsets of X with: { ∈ } (A) each Aα is compact; (B) Aα Aβ whenever α β; ⊂ N ≤ (C) X = Aα : α N . { ∈ } S Proposition, B. C., 1987 X Given X and Aα : α NN as above, if we deﬁne ϕ : NN 2 given by { ∈ } ∞ → ϕ(α) := A : β k = α k { β | | } then: k\=1 [ each ϕ(α) is countably compact (even more, all cluster points of any sequence in ϕ(α) remains in ϕ(α)). if ϕ(α) is compact then α ϕ(α) gives K-analytic structure to X . →

X has K-analytic structure if countably compact subsets=compact subsets.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Domination by Polish implies (many times) K-analyticity

Let X be a topological space Aα : α NN of subsets of X with: { ∈ } (A) each Aα is compact; (B) Aα Aβ whenever α β; ⊂ N ≤ (C) X = Aα : α N . { ∈ } Given α = (nk ) NN and m N, deﬁne S ∈ ∈ α m := (n1,n2,...,nm). | Proposition, B. C., 1987 X Given X and Aα : α NN as above, if we deﬁne ϕ : NN 2 given by { ∈ } ∞ → ϕ(α) := A : β k = α k { β | | } then: k\=1 [ each ϕ(α) is countably compact (even more, all cluster points of any sequence in ϕ(α) remains in ϕ(α)). if ϕ(α) is compact then α ϕ(α) gives K-analytic structure to X . → Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Domination by Polish implies (many times) K-analyticity

X has K-analytic structure if countably compact subsets=compact subsets. Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions The proof

Given α = (n1,n2,...,nm,...) we write

Cn n n := A : β k = α k 1, 2,..., k { β | | } [ Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions The proof Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions The proof Proposition, B. C., 1987

X Given X and Aα : α NN as above, if we deﬁne ϕ : NN 2 given by { ∈ } ∞ → ϕ(α) := A : β = α { β |k |k } k=1 then: \ [ each ϕ(α) is countably compact (even more, all cluster points of any sequence in ϕ(α) remains in ϕ(α)). if ϕ(α) is compact then α ϕ(α) gives K-analytic structure to X . →

X has K-analytic structure if countably compact subsets=compact subsets.

SO we have ﬁnally proved

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions The proof

The claim proved is the second part of (2) below: Keep in mind Proposition Let X be a metric space and ψ : X 2Y multi-valued . TFAE: → 1 ψ is usco;

2 ψ is compact valued + For every sequence xn x in X if → yn ψ(xn), n N then (yn)n has a cluster point y ψ(x). ∈ ∈ ∈ Proposition, B. C., 1987

X has K-analytic structure if countably compact subsets=compact subsets.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions The proof

The claim proved is the second part of (2) below: Keep in mind Proposition Let X be a metric space and ψ : X 2Y multi-valued . TFAE: → 1 ψ is usco;

2 ψ is compact valued + For every sequence xn x in X if → yn ψ(xn), n N then (yn)n has a cluster point y ψ(x). ∈ ∈ ∈ SO we have ﬁnally proved X has K-analytic structure if countably compact subsets=compact subsets.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions The proof

The claim proved is the second part of (2) below: Keep in mind Proposition Let X be a metric space and ψ : X 2Y multi-valued . TFAE: → 1 ψ is usco;

2 ψ is compact valued + For every sequence xn x in X if → yn ψ(xn), n N then (yn)n has a cluster point y ψ(x). ∈ ∈ ∈ SO we have ﬁnally proved

Proposition, B. C., 1987

The claim proved is the second part of (2) below: Keep in mind Proposition Let X be a metric space and ψ : X 2Y multi-valued . TFAE: → 1 ψ is usco;

2 ψ is compact valued + For every sequence xn x in X if → yn ψ(xn), n N then (yn)n has a cluster point y ψ(x). ∈ ∈ ∈ SO we have ﬁnally proved

Proposition, B. C., 1987

X has K-analytic structure if countably compact subsets=compact subsets. Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Talagrand’s solution to a conjecture Corson

Theorem, Talagrand 1975 Every WCG Banach space E is weakly Lindel¨of.

Proof.-

Fix W E absolutely convex w-compact with E = spanW . ⊂ Given α = (nk ) NN, ∈ 1 1 Aα := n1W + BE n2W + BE n1W + BE ... ∗∗ ∩ 2 ∗∗ ∩ ··· ∩ k ∗∗ ∩ Proposition (E,w) K-analytic (E,w) Lindel¨of. ⇒ ⇒ ♣ Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Fr´echet-Montelspaces

Theorem, Dieudonn´e1954 Every Fr´echet-Montelspace E is separable (in particular H (Ω) is separable).

Proof.-

Fix V1 V2 Vn ... a basis of closed neighborhoods of 0. ⊃ ⊃ ··· ⊃ Given α = (nk ) NN, ∈ ∞ Aα := nk Vk . k\=1 Aα : α NN fundamental family of bdd closed sets=compact; { ∈ } Proposition EK-analytic +metrizable E Lindel¨of+ metrizable E separable.⇒ ⇒ ⇒ ♣ U := ∞ Uk α aco k=1 nk Schwartz, 1964 S

Any Borel linear map from a separable Banach space into D0(Ω) is continuous. In particular, the Closed Graph Theorem holds for linear maps

T : D0(Ω) D0(Ω). →

Km Ω R sucesi´on exhaustiva de compactos % ⊂

E1 , E2 , , Em , , E → → · · · → → · · · → 1 2 m U1 U1 U1

∪. ∪. ∪ . . . . · . . . ∪ . m . Unm ∪1 § ¤ α = (nk)k U § n1 ¤ ∪ ¦ ∪ ¥ U 2 . ¦∪. ¥ § n2 ¤ . . ¦∪. ¥ .

N Uβ Uα si α β; U := Uα : α N neigh. basis of 0 en E. ⊂ ≤ { ∈ } Aα := U compact & Aα A , α β; α◦ ⊂ β ≤ N E 0 = Aα : α N and E 0 sub-metrizable E 0 K-analytic sub-metrizable∪{ ∈ E} analytic. ⇒ ⇒ 0

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions

D0(Ω) is analytic

Theorem, D0(Ω) is analytic. The strong dual of every inductive limit of Fr´echet-Montelspaces is analytic. U := ∞ Uk α aco k=1 nk Schwartz, 1964 S

Any Borel linear map from a separable Banach space into D0(Ω) is continuous. In particular, the Closed Graph Theorem holds for linear maps

T : D0(Ω) D0(Ω). →

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions

D0(Ω) is analytic

Theorem, D0(Ω) is analytic. The strong dual of every inductive limit of Fr´echet-Montelspaces is analytic.

Km Ω R sucesi´on exhaustiva de compactos % ⊂

E1 , E2 , , Em , , E → → · · · → → · · · → 1 2 m U1 U1 U1

∪. ∪. ∪ . . . . · . . . ∪ . m . Unm ∪1 § ¤ α = (nk)k U § n1 ¤ ∪ ¦ ∪ ¥ U 2 . ¦∪. ¥ § n2 ¤ . . ¦∪. ¥ . Schwartz, 1964

Any Borel linear map from a separable Banach space into D0(Ω) is continuous. In particular, the Closed Graph Theorem holds for linear maps

T : D0(Ω) D0(Ω). →

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions

D0(Ω) is analytic

Theorem, D0(Ω) is analytic. The strong dual of every inductive limit of Fr´echet-Montelspaces is analytic.

Km Ω R sucesi´on exhaustiva de compactos % ⊂

E1 , E2 , , Em , , E → → · · · → → · · · → 1 2 m U1 U1 U1

∪. ∪. ∪ . . . . · . . . ∪ . m . Unm ∪1 § ¤ α = (nk)k U § n1 ¤ ∪ ¦ ∪ ¥ U 2 . ¦∪. ¥ § n2 ¤ . . ¦∪. ¥ .

U := ∞ Uk α aco k=1 nk S N Uβ Uα si α β; U := Uα : α N neigh. basis of 0 en E. ⊂ ≤ { ∈ } Aα := U compact & Aα A , α β; α◦ ⊂ β ≤ N E 0 = Aα : α N and E 0 sub-metrizable E 0 K-analytic sub-metrizable∪{ ∈ E} analytic. ⇒ ⇒ 0 U := ∞ Uk α aco k=1 nk S

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions

D0(Ω) is analytic

Theorem, D0(Ω) is analytic. The strong dual of every inductive limit of Fr´echet-Montelspaces is analytic.

Km Ω R sucesi´on exhaustiva de compactos % ⊂

E1 , E2 , , Em , , E → → · · · → → · · · → 1 2 m U1 U1 U1

∪. ∪. ∪ . . . . · . . . ∪ . m . Unm ∪1 § ¤ α = (nk)k U § n1 ¤ ∪ ¦ ∪ ¥ U 2 . ¦∪. ¥ § n2 ¤ . . ¦∪. ¥ .

Schwartz, 1964 N Uβ Uα si α β; U := Uα : α N neigh. basis of 0 en E. Any Borel⊂ linear map≤ from a separable{ Banach∈ } space into D0(Ω) is continuous. In particular,Aα := U thecompact Closed Graph & Aα TheoremA , α holdsβ; for linear maps α◦ ⊂ β ≤ N E 0 = Aα : α N and E 0 sub-metrizable E 0 K-analytic ∪{ ∈ } T : D0(Ω) D0(Ω). ⇒ sub-metrizable E analytic. → ⇒ 0 Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Metrizability of compact sets (I)

N K compact space & Aα : α N subsets of (K K) ∆. We { ∈ } × \ write: (A) each Aα is compact;

(B) Aα A whenever α β; ⊂ β ≤ N (C) (K K) ∆ = Aα : α N . × \ { ∈ } S Theorem (Orihuela, B.C. 1987) (A) + (B) + (C) + (D) K is metrizable. ⇒ (D) For each compact set F (K K) ∆, there is Aα such that ⊂ × \ F Aα . ⊂ Ascoli (A) each Bα is ∞-bdd & closed & equicontinuous Bα is k k ⇒ ∞-compact; k k (B) Bα Bβ whenever α β; ⊂ ≤N (C) C(K) = Bα : α N . { ∈ } S

1 Given α NN, deﬁne Nα := (K K) Aα . ∈ × \ 2 Nα is a basis of open neighborhoods of ∆;

3 1 Bα := f C(K): f ∞ n1, f (x) f (y) m , whenever (x,y) Nα m ; m{ ∈ k k ≤ | − | ≤ ∈ | } for α := (nm,nm+1,...), m N. | ∈ 4

5 (C(K), ∞) is K-analytic +metrizable E Lindel¨of+ metrizable E separablek k K is metrizable. ⇒ ⇒ ⇒ ♣

K compact space & Aα : α NN subsets of (K K) ∆. We write: { ∈ } × \ (A) each Aα is compact; (B) Aα A whenever α β; ⊂ β ≤ (C) (K K) ∆ = Aα : α NN . × \ { ∈ } S

Proof.-

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Metrizability of compact sets (I)

Theorem (Orihuela, B.C. 1987)

(A) + (B) + (C) + (D) K is metrizable. ⇒ (D) For each compact set F (K K) ∆, there is Aα such that F Aα . ⊂ × \ ⊂ Ascoli (A) each Bα is ∞-bdd & closed & equicontinuous Bα is k k ⇒ ∞-compact; k k (B) Bα Bβ whenever α β; ⊂ ≤N (C) C(K) = Bα : α N . { ∈ } S

1 Given α NN, deﬁne Nα := (K K) Aα . ∈ × \ 2 Nα is a basis of open neighborhoods of ∆;

3 1 Bα := f C(K): f ∞ n1, f (x) f (y) m , whenever (x,y) Nα m ; m{ ∈ k k ≤ | − | ≤ ∈ | } for α := (nm,nm+1,...), m N. | ∈ 4

5 (C(K), ∞) is K-analytic +metrizable E Lindel¨of+ metrizable E separablek k K is metrizable. ⇒ ⇒ ⇒ ♣

Proof.-

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions K compact space & Aα : α NN subsets of (K K) ∆. We write: { ∈ } × \ (A) each Aα is compact; Metrizability(B) Aα A whenever of compactα β; sets (I) ⊂ β ≤ (C) (K K) ∆ = Aα : α NN . × \ { ∈ } S Theorem (Orihuela, B.C. 1987)

2 Nα is a basis of open neighborhoods of ∆;

3 1 Bα := f C(K): f ∞ n1, f (x) f (y) m , whenever (x,y) Nα m ; m{ ∈ k k ≤ | − | ≤ ∈ | } for α := (nm,nm+1,...), m N. | ∈ 4

5 (C(K), ∞) is K-analytic +metrizable E Lindel¨of+ metrizable E separablek k K is metrizable. ⇒ ⇒ ⇒ ♣

(A) + (B) + (C) + (D) K is metrizable. ⇒ (D) For each compact set F (K K) ∆, there is Aα such that F Aα . ⊂ × \ ⊂ Proof.-

1 Given α NN, deﬁne Nα := (K K) Aα . ∈ × \ Ascoli (A) each Bα is ∞-bdd & closed & equicontinuous Bα is k k ⇒ ∞-compact; k k (B) Bα Bβ whenever α β; ⊂ ≤N (C) C(K) = Bα : α N . { ∈ } S

3 1 Bα := f C(K): f ∞ n1, f (x) f (y) m , whenever (x,y) Nα m ; m{ ∈ k k ≤ | − | ≤ ∈ | } for α := (nm,nm+1,...), m N. | ∈ 4

5 (C(K), ∞) is K-analytic +metrizable E Lindel¨of+ metrizable E separablek k K is metrizable. ⇒ ⇒ ⇒ ♣

(A) + (B) + (C) + (D) K is metrizable. ⇒ (D) For each compact set F (K K) ∆, there is Aα such that F Aα . ⊂ × \ ⊂ Proof.-

1 Given α NN, deﬁne Nα := (K K) Aα . ∈ × \ 2 Nα is a basis of open neighborhoods of ∆; Ascoli (A) each Bα is ∞-bdd & closed & equicontinuous Bα is k k ⇒ ∞-compact; k k (B) Bα Bβ whenever α β; ⊂ ≤N (C) C(K) = Bα : α N . { ∈ } S

5 (C(K), ∞) is K-analytic +metrizable E Lindel¨of+ metrizable E separablek k K is metrizable. ⇒ ⇒ ⇒ ♣

(A) + (B) + (C) + (D) K is metrizable. ⇒ (D) For each compact set F (K K) ∆, there is Aα such that F Aα . ⊂ × \ ⊂ Proof.-

1 Given α NN, deﬁne Nα := (K K) Aα . ∈ × \ 2 Nα is a basis of open neighborhoods of ∆;

3 1 Bα := f C(K): f ∞ n1, f (x) f (y) m , whenever (x,y) Nα m ; m{ ∈ k k ≤ | − | ≤ ∈ | } for α := (nm,nm+1,...), m N. | ∈ (B) Bα Bβ whenever α β; ⊂ ≤N (C) C(K) = Bα : α N . { ∈ } 5 (C(K), ∞) isSK-analytic +metrizable E Lindel¨of+ metrizable E separablek k K is metrizable. ⇒ ⇒ ⇒ ♣

(A) + (B) + (C) + (D) K is metrizable. ⇒ (D) For each compact set F (K K) ∆, there is Aα such that F Aα . ⊂ × \ ⊂ Proof.-

1 Given α NN, deﬁne Nα := (K K) Aα . ∈ × \ 2 Nα is a basis of open neighborhoods of ∆;

3 1 Bα := f C(K): f ∞ n1, f (x) f (y) m , whenever (x,y) Nα m ; m{ ∈ k k ≤ | − | ≤ ∈ | } for α := (nm,nm+1,...), m N. | ∈ 4 Ascoli (A) each Bα is ∞-bdd & closed & equicontinuous Bα is k k ⇒ ∞-compact; k k N (C) C(K) = Bα : α N . { ∈ } 5 (C(K), ∞) isSK-analytic +metrizable E Lindel¨of+ metrizable E separablek k K is metrizable. ⇒ ⇒ ⇒ ♣

(A) + (B) + (C) + (D) K is metrizable. ⇒ (D) For each compact set F (K K) ∆, there is Aα such that F Aα . ⊂ × \ ⊂ Proof.-

1 Given α NN, deﬁne Nα := (K K) Aα . ∈ × \ 2 Nα is a basis of open neighborhoods of ∆;

(A) + (B) + (C) + (D) K is metrizable. ⇒ (D) For each compact set F (K K) ∆, there is Aα such that F Aα . ⊂ × \ ⊂ Proof.-

1 Given α NN, deﬁne Nα := (K K) Aα . ∈ × \ 2 Nα is a basis of open neighborhoods of ∆;

3 1 Bα := f C(K): f ∞ n1, f (x) f (y) m , whenever (x,y) Nα m ; m{ ∈ k k ≤ | − | ≤ ∈ | } for α := (nm,nm+1,...), m N. | ∈ 4 Ascoli (A) each Bα is ∞-bdd & closed & equicontinuous Bα is k k ⇒ ∞-compact; k k (B) Bα Bβ whenever α β; ⊂ ≤N (C) C(K) = Bα : α N . { ∈ } S Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions K compact space & Aα : α NN subsets of (K K) ∆. We write: { ∈ } × \ (A) each Aα is compact; Metrizability(B) Aα A whenever of compactα β; sets (I) ⊂ β ≤ (C) (K K) ∆ = Aα : α NN . × \ { ∈ } S Theorem (Orihuela, B.C. 1987)

(A) + (B) + (C) + (D) K is metrizable. ⇒ (D) For each compact set F (K K) ∆, there is Aα such that F Aα . ⊂ × \ ⊂ Proof.-

1 Given α NN, deﬁne Nα := (K K) Aα . ∈ × \ 2 Nα is a basis of open neighborhoods of ∆;

3 1 Bα := f C(K): f ∞ n1, f (x) f (y) m , whenever (x,y) Nα m ; m{ ∈ k k ≤ | − | ≤ ∈ | } for α := (nm,nm+1,...), m N. | ∈ 4 Ascoli (A) each Bα is ∞-bdd & closed & equicontinuous Bα is k k ⇒ ∞-compact; k k (B) Bα Bβ whenever α β; ⊂ ≤N (C) C(K) = Bα : α N . { ∈ } 5 (C(K), ∞) isSK-analytic +metrizable E Lindel¨of+ metrizable E separablek k K is metrizable. ⇒ ⇒ ⇒ ♣ Theorem (Orihuela, B.C. 1987) (K,U) a compact uniform space with a basis for the uniformity BU = Nα : α NN satisfying: ∈ N Nβ Nα si α β whenever α,β N . ⊂ ≤ ∈ Then K is metrizable.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Metrizability of compact sets (I): A diﬀerent formulation

We didn’t stated our result below as presented.

K compact space & Aα : α NN subsets of (K K) ∆. We write: { ∈ } × \ (A) each Aα is compact;

(B) Aα A whenever α β; ⊂ β ≤ (C) (K K) ∆ = Aα : α NN . × \ { ∈ } Theorem (Orihuela, B.C.S 1987) (A) + (B) + (C) + (D) K is metrizable. ⇒ (D) For each compact set F (K K) ∆, there is Aα such that F Aα . ⊂ × \ ⊂ Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Metrizability of compact sets (I): A diﬀerent formulation

We didn’t stated our result below as presented.

K compact space & Aα : α NN subsets of (K K) ∆. We write: { ∈ } × \ (A) each Aα is compact;

Theorem (Orihuela, B.C. 1987) (K,U) a compact uniform space with a basis for the uniformity BU = Nα : α NN satisfying: ∈ N Nβ Nα si α β whenever α,β N . ⊂ ≤ ∈ Then K is metrizable. Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions The original paper

Math. Z. 195, 365-381 (1987) Mathematische Zeitschrift 9 Springer-Verlag 1987

On Compactness in Locally Convex Spaces

B. Cascales and J. Orihuela Departamento de Analisis Matematico, Facultad de Matematicas, Universidad de Murcia, E-30.001-Murcia-Spain

1. Introduction and Terminology

The purpose of this paper is to show that the behaviour of compact subsets in many of the locally convex spaces that usually appear in Functional Analysis is as good as the corresponding behaviour of compact subsets in Banach spaces. Our results can be intuitively formulated in the following terms: Deal- ing with metrizable spaces or their strong duals, and carrying out any of the usual operations of countable type with them, we ever obtain spaces with their precompact subsets metrizable, and they even give good performance for the weak topology, indeed they are weakly angelic, [-14], and their weakly compact subsets are metrizable if and only if they are separable. The first attempt to clarify the sequential behaviour of the weakly compact subsets in a Banach space was made by V.L. Smulian [26] and R.S. Phillips [23]. Their results are based on an argument of metrizability of weakly compact subsets (see Floret [14], pp. 29-30). Smulian showed in [26] that a relatively compact subset is relatively sequentially compact for the weak topology of a Banach space. He also proved that the concepts of relatively countably compact and relatively sequentially compact coincide if the weak-* dual is separable. The last result was extended by J. Dieudonn6 and L. Schwartz in [-9] to submetrizable locally convex spaces. The converse to Smulian's theorem was stated by W.F. Eberlein [10]. This result was extended by A. Grothendieck [-15], to spaces of continuous functions on compact spaces endowed with the pointwise convergence topology. A combination of results by A. Grothendieck, D.H. Fremlin, J.D. Pryce and M. DeWilde allow K. Floret [14], p. 36, to give a proof of a general version for the Eberlein-Smulian theorem. In spite of its powerful applications, the scope of the "Eberlein- Smulian theorem does not include some important classes of locally convex spaces and it gives no information about the metrizability of the compact subsets. Dealing with compactness in a locally convex space E two questions appear to arise: (1) Are the compact subsets of E metrizable? (2) Is the space E weakly angelic? 378 B. Cascales and J. Orihuela

Corollary 1.15. Let El3;] be a separable LCS of the class (5. Then E'[3;p~(E', E)] is analytic and in particular E'[o-(E', E)] is analytic. Proof The separability of E implies that E'[3;pc(E',E)] is submetrizable. The (5-representation of E in E' gives us a family {A/ e~N N} of countable equicontinuous subsets of E' such that A~ ~ A~ whenever ~ < fl in N ~s. Every A~ is relatively countably compact in E'113;pc(E',E)]. The angelic character of E'[3;pc(E',E)] provides us with the situation of Theorem 15, hence the result. Q.E.D.

The former corollary contains as particular cases some of the results proved by M. DeWilde in [8], VII.2, and of course new cases of applications because of the good stability properties of the class (5.

6. Regularity Conditions in Inductive Limits

The regularity and retractivity conditions have been studied by different authors: K. Floret introduces the notion of sequentially retractive inductive limit in [12], Bierstedt and Meise introduce the compact-regular inductive limits in [3] and H. Neus gives the definition of sequentially compact-regular inductive limit in [20] as well as deep results on the equivalence between the different notions of regularity and retractivity for inductive limits of sequences of normed spaces. In 1128], M. Valdivia has studied the former properties and other related ones for the weak topology of inductive limits of normed spaces. Recently, the authors have proved in [-6] that for an inductive limit of an increasing sequence of metrizable LCS the conditions of sequentially retractive, Proto Idea sequentiallyStarting Point compact-regular,Domination by Polish&Appl. compact-regularDomination by and second precompactly countable Extensions&Open retractive questions [6], are equivalent. Other results in this context for generalized inductive limits A nicehave applications been obtained by the first author in [-4]. The purpose of this short paragraph is to obtain the former equivalences in the general case of inductive limits of spaces of the class (5, and thus - in some way - give a certain answer to K. Floret, who expounds in [13] that it is desirable to study this kind of results in general cases. Theorem 16. If E [3;] = limE, J3;,] is an inductive limit of an increasing sequence of subspaces E,[-3;J belonging to the class (5, then the following statements are equivalent: (i) E[3;] is sequentially retractive. (ii) E [3;] is sequentially compact-regular. (iii) E[3;] is compact-regular. (iv) E113;] is precompaetly retractive. If every E,[-3;,] is complete, the former conditions are also equivalent to the following: (v) For every precompact subset A of E[-3;] there is a positive integer n such that A is contained in E,113;,3 and it is precompact in this space. Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions The techniques seems to be useful yet

The results have been used by many authors over the years: Bonet, Dierolf,MR: Selected Maestre, Matches Bistrom, for: Citations Robertson, of 895307 Valdivia, Wengeroth,http://www.ams.org/mathscinet/search/publications.html?foo... Lindstrom, Bierstedt, etc.

MR2666299 (Review) Ferrando, J. C. ; Ka̧kol, Jerzy ; López Pellicer, M. ; Śliwa, W. Web-compact spaces, Fréchet-Urysohn groups and a Suslin closed graph theorem. Math. Nachr. 283 (2010), no. 5, 704--711.

MR2596470 (2011b:46006) Albanese, Angela A. ; Bonet, José ; Ricker, Werner J. Grothendieck spaces with the Dunford-Pettis property. Positivity 14 (2010), no. 1, 145--164.

MR2541044 (2010e:46005) Ka̧kol, J. ; López Pellicer, M. ; Todd, A. R. A topological vector space is Fréchet-Urysohn if and only if it has bounded tightness. Bull. Belg. Math. Soc. Simon Stevin 16 (2009), no. 2, 313--317.

MR2346899 (2009g:46007) Drewnowski, Lech . Resolutions of topological linear spaces and continuity of linear maps. J. Math. Anal. Appl. 335 (2007), no. 2, 1177--1194.

MR2333802 (2008h:54018) Tkachuk, V. V. A selection of recent results and problems in C p-theory. Topology Appl. 154 (2007), no. 12, 2465--2493.

MR2150789 (2006e:54007) Tkachuk, V. V. A space C p (X) is dominated by irrationals if and only if it is K -analytic. Acta Math. Hungar. 107 (2005), no. 4, 253--265.

Mirror Sites Providence, RI USA

American Mathematical Society 201 Charles Street Providence, RI 02904-2294

1 of 1 2/18/11 8:29 AM Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Applications

INDEX 5 Strongly web-compact spaces and Closed Graph Theorem 167 5.1 Strongly web-compact spaces ...... 167 5.2 Products of strongly web-compact spaces ...... 168 5.3 A Closed Graph Theorem for strongly web-compact spaces .. 170

6 Weakly analytic spaces 175 6.1 Something about analytic spaces ...... 175 6.2 Christensen theorem ...... 182 6.3 Subspaces of analytic spaces ...... 189 6.4 Trans-separable topological spaces ...... 192 6.5 Weakly analytic spaces need not be analytic ...... 200 6.6 When a weakly analytic locally convex space is analytic? ... 204 6.7 Weakly compact density condition ...... 206 6.8 More examples of non-separable weakly analytic tvs ...... 215

7 K-analytic Baire spaces 223 Descriptive Topology in Selected 7.1 Baire tvs with a bounded resolution ...... 223 Topics of Functional Analysis 7.2 Continuous maps on spaces with resolutions ...... 229

J. K¸akol, W. Kubi´s, M. Lopez Pellicer 8 A three-space property for analytic spaces 235 8.1 Corson’s example ...... 235 December 29, 2009 8.2 A positive result and a counterexample ...... 239

9 K-analytic and analytic spaces Cp(X) 245 9.1 Talagrand’s theorem for spaces Cp(X) ...... 245 9.2 Christensen and Calbrix theorem for Cp(X) ...... 249 9.3 Bounded resolutions for spaces Cp(X) ...... 257 9.4 More examples of K-analytic spaces Cp(X) and Cp(X,E) ... 278 9.5 K-analytic spaces Cp(X) over locally compact groups X .... 280

9.6 K-analytic group Xp∧ of homomorphisms ...... 284

10 Precompact sets in (LM)-spaces and dual metric spaces 289 10.1 The case of (LM)-spaces, elementary approach ...... 289 10.2 The case of dual metric spaces, elementary approach ..... 292

11 Metrizability of compact sets in class G 295 11.1 Spaces in class G and examples ...... 295 11.2 Cascales-Orihuela theorem and applications ...... 298

4 Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Applications

INDEX 12 Weakly and ∗-weakly realcompact locally convex spaces 305 12.1 Tightness and quasi-Suslin weak duals ...... 305 12.2 A Kaplansky type theorem about tightness ...... 309 12.3 More about K-analytic spaces in class G ...... 314 12.4 Every (WCG) Fr´echet space is weakly K-analytic ...... 318 12.5 About a theorem of Amir-Lindenstrauss and (CSPP) property 325 12.6 An example of R. Pol ...... 331 12.7 One more fact about Banach spaces Cc(X) over compact scattered X ...... 337

13 Corson property (C) and tightness 341 13.1 Property (C) and weakly Lindel¨of Banach spaces ...... 341 13.2 Property (C) for Banach spaces C(K) ...... 348

14 Fréchet-Urysohn spaces and topological groups 353 14.1 Fréchet-Urysohn topological spaces ...... 353 Descriptive Topology in Selected 14.2 A few facts about Fréchet-Urysohn topological groups ..... 356 Topics of Functional Analysis 14.3 Sequentially complete Fréchet-Urysohn lcs are Baire ...... 362

14.4 Three-space property for spaces Fr´echet-Urysohn ...... 366 J. K¸akol, W. Kubi´s, M. Lopez Pellicer 14.5 Topological vector spaces with bounded tightness ...... 369 December 29, 2009 15 Sequential conditions in class G 373 15.1 Fr´echet-Urysohn lcs are metrizable in class G ...... 373 15.2 Sequential (LM)-spaces and dual metric spaces ...... 380

15.3 (LF )-spaces with property C3− and ϕ ...... 392

16 Tightness and distinguished Fr´echet spaces 401 16.1 A characterization of distinguished spaces in term of tightness 401 16.2 G-bases and tightness ...... 409 16.3 G-bases, bounding and dominating cardinals and tightness .. 414 16.4 More about the Wulbert-Morris space Cc(ω1) ...... 427

17 Banach spaces with many projections 433 17.1 Preliminaries, model-theoretic tools ...... 433 17.2 Projections from elementary submodels ...... 441 17.3 Lindel¨ofproperty of weak topologies ...... 444 17.4 Separable complementation property ...... 445 17.5 Projectional skeletons ...... 450

5 Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Tkachuk’s views about the metrizability result Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions We came back 25 years later

Topology and its Applications 158 (2011) 204–214

Contents lists available at ScienceDirect

Topology and its Applications

www.elsevier.com/locate/topol

Domination by second countable spaces and Lindelöf Σ-property

a,1,2 a,1,2 b, ,3,4 B. Cascales ,J.Orihuela , V.V. Tkachuk ∗

a Departamento de Matemáticas, Facultad de Ciencias, Universidad de Murcia, 30.100, Espinardo, Murcia, Spain b Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco, 186, Col. Vicentina, Iztapalapa, C.P. 09340, México D.F., Mexico

article info abstract

Article history: Given a space M,afamilyofsets of a space X is ordered by M if {A K : K is a Received 19 August 2010 A A = compact subset of M}andK L implies A K AL .Westudytheclass of spaces which Accepted 28 October 2010 ⊂ ⊂ M have compact covers ordered by a second countable space. We prove that a space C p (X) belongs to if and only if it is a Lindelöf Σ-space. Under MA(ω ),ifX is compact and Keywords: M 1 (X X) # has a compact cover ordered by a Polish space then X is metrizable; here (Strong) domination by irrationals × \ # (x, x): x X is the diagonal of the space X.Besides,ifX is a compact space of (Strong) domination by a second countable ={ ∈ } countable tightness and X2 # belongs to then X is metrizable in ZFC. space \ M Diagonal We also consider the class of spaces X which have a compact cover ordered by M∗ F Metrization a second countable space with the additional property that, for every compact set P X ⊂ Orderings by irrationals there exists F with P F . It is a ZFC result that if X is a compact space and (X X) # Orderings by a second countable space ∈ F ⊂ × \ belongs to ∗ then X is metrizable. We also establish that, under CH, if X is compact and Compact cover M C p (X) belongs to ∗ then X is countable. Function spaces M  2010 Elsevier B.V. All rights reserved. Cosmic spaces -spaces ℵ0 Lindelöf Σ-space Compact space Metrizable space

0. Introduction

Given a space X we denote by (X) the family of all compact subsets of X.Oneofaboutadozenequivalentdeﬁnitions says that X is a Lindelöf Σ-space (orK has the Lindelöf Σ-property) if there exists a second countable space M and a compact- valued upper semicontinuous map ϕ M X such that ϕ(x): x M X (see, e.g., [23, Section 5.1]). It is worth : → { ∈ }= mentioning that in Functional Analysis, the same concept is usually referred to as a countably K -determined space. Suppose that X is a Lindelöf Σ-space and hence we can! ﬁnd a compact-valued upper semicontinuous surjective map ϕ M X for some second countable space M.IfweletF ϕ(x): x K for any compact set K M then the family : → K = { ∈ } ⊂ F : K (M) consists of compact subsets of X,coversX and K L implies F F .Wewillsaythat is an M- ={ K ∈ } ⊂ K ⊂ L orderedF compactK cover of X. ! F The class of spaces with an M-ordered compact cover for some second countable space M,wasintroducedby Cascales and OrihuelaM in [9]. They proved, among other things, that a Dieudonné complete space is Lindelöf Σ if and only

* Corresponding author. E-mail addresses: [email protected] (B. Cascales), [email protected] (J. Orihuela), [email protected] (V.V. Tkachuk). 1 Research supported by FEDER and MEC, Project MTM2008-05396. 2 Research supported by Fundación Séneca de la CARM, Project 08848/PI/08. 3 Research supported by Consejo Nacional de Ciencia y Tecnología de México, Grant U48602-F. 4 Research supported by Programa Integral de Fortalecimiento Institucional (PIFI), Grant 34536-55.

208 B. Cascales et al. / Topology and its Applications 158 (2011) 204–214

Proof. Fix any set X Z and assume that X is dominated by a second countable space. For any set A X we denote by ⊂ ⊂ clX (A) (or clZ (A))theclosureofthesetA in the space X (or in Z respectively). By Proposition 2.6, there exist a cover of the space X and a countable network with respect to such that for every C and any countable A C the setC N C ∈ C ⊂ clX (A) is compact and contained in C. If C and C is not closed in Z then we can ﬁnd a point x cl (C) C.BycountabletightnessofZ, there exists ∈ ∈ Z \ acountableC A C such that x cl (A).ThesetF cl (A) C is compact and hence closed in Z; as a consequence, ⊂ ∈ Z = X ⊂ x cl (A) F C. This contradiction shows that every C is compact being closed in X.Thus is a countable network ∈ Z ⊂ ⊂ ∈ with respect to the compact cover of the space X, i.e., X hasC the Lindelöf Σ-property. N C 2 2.9. Theorem. If X is a compact space with t(X) ω and X2 # is dominated by a second countable space then X is metrizable. ! \

Proof. The space X2 also has countable tightness [1, Theorem 2.3.3] so we can apply Theorem 2.8 to the set X 2 # X X 2 \ ⊂ × to conclude that X # is a Lindelöf Σ-space; this easily implies that the diagonal # is a Gδ-subset of X X and hence X \ × is metrizable by [11, 3.12.22(e)]. 2 2.10. Corollary. If X is a Corson compact space or a ﬁrst countable compact space such that X 2 # is dominated by a second countable space then X is metrizable. \

2.11. Theorem. If X is a dyadic compact space and X 2 # is dominated by a second countable space then X is metrizable. \

Proof. If X is first countable then it is metrizable by [11, 3.12.12(e)]. Therefore we can assume that there exists a point x X of uncountable character in X.Apply[11,3.12.12(i)]tofindanuncountableone-pointcompactificationA of a discrete ∈ space such that A X and x is the unique non-isolated point of A.ThenB (A x ) x is an uncountable closed discrete ⊂ = \{ } × { } subspace of (X X) # while we have ext(X2 #) ω by Theorem 2.1(h), a contradiction. × \ \ = ψ(α) ψ(α), (closure in K K) 2 The above results show that, to prove that any⊂ compact space X with ×X2 # dominated by a second countable space is take x ψ(α); \ metrizable, it∈ suffices to show that any such space has a countable tightness. While we don’t know whether this implication is truethere in general, is A weψ( doα) present countable some with partialx progressA; in this direction. ⊂ ∈ is x A x ψ(α); 2 2.12. Theorem. Assume MA(ω1) and suppose that X is a compact space such that X # is P-dominated. Then X has a small diagonal ∈ ⇒ ∈ \ and henceotherwise, t(X) ωx. (A A) x is cluster point of a sequence in ψ(α) x ψ(α). = ∈ \ ⇒ ⇒ ∈ 2 Proof. Suppose that A zα: α < ω1 X # and α β implies zα zβ .FixaP-directed cover K p: p P of compact Proposition, B.2 C., 1987={ } ⊂ \ $= $= { ∈ } subsets of X #.Takepα P such that zα K pα for any α < ω1. \ ∈ ∈ It follows from MA( ) that there exists p such that p p for any < .ThesetP K : q and q p ω1N P N α !∗(K K) ∆ α ω1 q P ∗ Given X and Aα : α N as above, if we define∈ ψ : N 2 × \ given by = { ∈ = } is σ -compact and A P .Consequently,thereisq P for which Kq A is uncountable; therefore the set Kq A witnesses { ⊂∈ } ∈ → ∩ ! ∩ the small diagonal property of X.Sincenospacewithasmalldiagonalcanhaveaconvergent∞ ω1-sequence, it follows from [16, Theorem 1.2] that X has no free sequences of length ω , i.e., t(X) ω. ψ(α) := 1 Aβ : β m!= α m { | | 2} m\=1 [ 2.13. Corollary. Under MA(ω ), if X is a compact space such that X2 # is dominated by a Polish space then X is metrizable. then: 1 \ Proof. Apply Proposition 2.2 to see that the space X2 # is dominated by P so t(X) ω by Theorem 2.12 and hence X is each ψ(α) is countably compact (even more,\ all cluster points of any sequence! in ψ(α) remains in ψ(α)). metrizable by Theorem 2.9. if ψ(α) is compact then α ψ(α) gives K-analytic structure to (K K) ∆. 2 → × \ In the rest of this section we study the spaces hereditarily dominated by a second countable space. The motivation here is a result of Hodel established in [14, Corollary 4.13]; it says that any hereditarily Lindelöf Σ-space is cosmic. We will look at this hereditary property in function spaces to show that a somewhat stronger statement is true in a general situation under Martin’s Axiom. 1 The(A) following+ fact(B) is an+ immediate(C) + consequence MA(ω1 of) [26, PropositionK has 2.7]. small diagonal, i.e., for any uncountable set A (K K) ⇒∆ there exists an uncountable B A such 2.14. Proposition. If X is a space which has a countable network modulo a cover of X by countably compact sets then C p (X) is Lindelöf ⊂ × \ X ⊂ Σ-framed,that i.e.,B there is∆ a Lindelöf = /0.Σ-space L such that C p(X) L R . ∩ ⊂ ⊂ 2.15.2 Theorem.K hasA smallspace C p(X diagonal) is dominated by a secondK has countable countable space if and onlytightness if it is Lindelöf Σ. K K has ⇒ ⇒ × Proof.countableWe must only prove tightness; necessity. Suppose that C p(X) is dominated by a second countable space M and fix a family F : K (M) which witnesses this. It follows from Proposition 2.14 and Proposition 2.6 that C (C (X)) is Lindelöf Σ- { K ∈ K } p p framed.3 (K ApplyingK) [21, Theorem∆ is K 3.5]-analytic we conclude that υ(C p (X)) is a Lindelöf Σ-space and hence υ X is a Lindelöf Σ-space by [21, Corollary× 3.6].\

Proof.-

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Metrizability of compact sets (II)

K compact space & Aα : α NN subsets of (K K) ∆. We write: { ∈ } × \ (A) each Aα is compact;

(B) Aα A whenever α β; ⊂ β ≤ (C) (K K) ∆ = Aα : α NN . × \ { ∈ } S Theorem (Orihuela, Tkachuk, B.C. 2011)

(A) + (B) + (C) + MA(ω1) K is metrizable. ⇒ (K K) ∆ is Lindel¨of ∆ is Gδ K is metrizable. ⇒ × \ ⇒ ⇒ ♣

208 B. Cascales et al. / Topology and its Applications 158 (2011) 204–214

2.11. Theorem. If X is a dyadic compact space and X 2 # is dominated by a second countable space then X is metrizable. \

Proof. If X is first countable then it is metrizable by [11, 3.12.12(e)]. Therefore we can assume that there exists a point x X of uncountable character in X.Apply[11,3.12.12(i)]tofindanuncountableone-pointcompactificationA of a discrete ∈ space such that A X and x is the unique non-isolated point of A.ThenB (A x ) x is an uncountable closed discrete ⊂ = \{ } × { } subspace of (X X) # while we have ext(X2 #) ω by Theorem 2.1(h), a contradiction. × \ \ = ψ(α) ψ(α), (closure in K K) 2 The above results show that, to prove that any⊂ compact space X with ×X2 # dominated by a second countable space is take x ψ(α); \ metrizable, it∈ suffices to show that any such space has a countable tightness. While we don’t know whether this implication is truethere in general, is A weψ( doα) present countable some with partialx progressA; in this direction. ⊂ ∈ is x A x ψ(α); 2 2.12. Theorem. Assume MA(ω1) and suppose that X is a compact space such that X # is P-dominated. Then X has a small diagonal ∈ ⇒ ∈ \ and henceotherwise, t(X) ωx. (A A) x is cluster point of a sequence in ψ(α) x ψ(α). = ∈ \ ⇒ ⇒ ∈ 2 Proof. Suppose that A zα: α < ω1 X # and α β implies zα zβ .FixaP-directed cover K p: p P of compact Proposition, B.2 C., 1987={ } ⊂ \ $= $= { ∈ } subsets of X #.Takepα P such that zα K pα for any α < ω1. \ ∈ ∈ It follows from MA( ) that there exists p such that p p for any < .ThesetP K : q and q p ω1N P N α !∗(K K) ∆ α ω1 q P ∗ Given X and Aα : α N as above, if we define∈ ψ : N 2 × \ given by = { ∈ = } is σ -compact and A P .Consequently,thereisq P for which Kq A is uncountable; therefore the set Kq A witnesses { ⊂∈ } ∈ → ∩ ! ∩ the small diagonal property of X.Sincenospacewithasmalldiagonalcanhaveaconvergent∞ ω1-sequence, it follows from [16, Theorem 1.2] that X has no free sequences of length ω , i.e., t(X) ω. ψ(α) := 1 Aβ : β m!= α m { | | 2} m\=1 [ 2.13. Corollary. Under MA(ω ), if X is a compact space such that X2 # is dominated by a Polish space then X is metrizable. then: 1 \ Proof. Apply Proposition 2.2 to see that the space X2 # is dominated by P so t(X) ω by Theorem 2.12 and hence X is each ψ(α) is countably compact (even more,\ all cluster points of any sequence! in ψ(α) remains in ψ(α)). metrizable by Theorem 2.9. if ψ(α) is compact then α ψ(α) gives K-analytic structure to (K K) ∆. 2 → × \ In the rest of this section we study the spaces hereditarily dominated by a second countable space. The motivation here is a result of Hodel established in [14, Corollary 4.13]; it says that any hereditarily Lindelöf Σ-space is cosmic. We will look at this hereditary property in function spaces to show that a somewhat stronger statement is true in a general situation under Martin’s Axiom. The following fact is an immediate consequence of [26, Proposition 2.7].

2.14. Proposition. If X is a space which has a countable network modulo a cover of X by countably compact sets then C p (X) is Lindelöf X Σ-framed, i.e., there is a Lindelöf Σ-space L such that C p(X) L R . ⊂ ⊂

2.15.2 Theorem.K hasA smallspace C p(X diagonal) is dominated by a secondK has countable countable space if and onlytightness if it is Lindelöf Σ. K K has ⇒ ⇒ × Proof.countableWe must only prove tightness; necessity. Suppose that C p(X) is dominated by a second countable space M and ﬁx a family F : K (M) which witnesses this. It follows from Proposition 2.14 and Proposition 2.6 that C (C (X)) is Lindelöf Σ- { K ∈ K } p p framed.3 (K ApplyingK) [21, Theorem∆ is K 3.5]-analytic we conclude that υ(C p (X)) is a Lindelöf Σ-space and hence υ X is a Lindelöf Σ-space by [21, Corollary× 3.6].\

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Metrizability of compact sets (II)

K compact space & Aα : α NN subsets of (K K) ∆. We write: { ∈ } × \ (A) each Aα is compact;

(B) Aα A whenever α β; ⊂ β ≤ (C) (K K) ∆ = Aα : α NN . × \ { ∈ } S Theorem (Orihuela, Tkachuk, B.C. 2011)

(A) + (B) + (C) + MA(ω1) K is metrizable. ⇒ Proof.-

1 (A) + (B) + (C) + MA(ω1) K has small diagonal, i.e., for any uncountable set A (K K) ⇒∆ there exists an uncountable B A such ⊂ × \ ⊂ that B ∆ = /0. ∩ (K K) ∆ is Lindel¨of ∆ is Gδ K is metrizable. ⇒ × \ ⇒ ⇒ ♣

ψ(α) ψ(α), (closure in K K) ⊂ × take x ψ(α); ∈ there is A ψ(α) countable with x A; ⊂ ∈ is x A x ψ(α); ∈ ⇒ ∈ otherwise, x (A A) x is cluster point of a sequence in ψ(α) x ψ(α). ∈ \ ⇒ ⇒ ∈ Proposition, B. C., 1987

(K K) ∆ Given X and Aα : α NN as above, if we deﬁne ψ : NN 2 given by { ∈ } → × \ ∞ ψ(α) := A : β m = α m { β | | } m\=1 [ then: each ψ(α) is countably compact (even more, all cluster points of any sequence in ψ(α) remains in ψ(α)). if ψ(α) is compact then α ψ(α) gives K-analytic structure to (K K) ∆. → × \

3 (K K) ∆ is K-analytic × \

208 B. Cascales et al. / Topology and its Applications 158 (2011) 204–214

2.11. Theorem. If X is a dyadic compact space and X 2 # is dominated by a second countable space then X is metrizable. \

Proof. If X is first countable then it is metrizable by [11, 3.12.12(e)]. Therefore we can assume that there exists a point x X of uncountable character in X.Apply[11,3.12.12(i)]tofindanuncountableone-pointcompactificationA of a discrete Proto Idea ∈ Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions space such that A X and x is the unique non-isolated point of A.ThenB (A x ) x is an uncountable closed discrete ⊂ = \{ } × { } subspace of (X X) # while we have ext(X2 #) ω by Theorem 2.1(h), a contradiction. × \ \ = 2 The above results show that, to prove that any compact space X with X2 # dominated by a second countable space is Metrizability of compact sets (II) \ metrizable, it suffices to show that any such space has a countable tightness. While we don’t know whether this implication is true in general, we do present some partial progress in this direction.

2 2.12. Theorem. Assume MA(ω1) and suppose that X is a compact space such that X # is P-dominated. Then X has a small diagonal N \ K andcompact hence t(X) spaceω. & Aα : α N subsets of (K K) ∆. We write: = { ∈ } × \ (A) each A is compact; 2 Proof. Suppose thatα A zα: α < ω1 X # and α β implies zα zβ .FixaP-directed cover K p: p P of compact 2 ={ } ⊂ \ $= $= { ∈ } subsets of X #.Takepα P such that zα K pα for any α < ω1. \ ∈ ∈ (B)It followsA fromA MAwhenever(ω1) that thereα exists pβ;P such that pα ∗ p for any α < ω1.ThesetP Kq: q P and q ∗ p α β ∈ ! = { ∈ = } is σ -compact⊂ and A P .Consequently,thereis≤ q P for which Kq A is uncountable; therefore the set Kq A witnesses ⊂ ∈ ∩ ∩ the small diagonal property of X.Sincenospacewithasmalldiagonalcanhaveaconvergentω -sequence,! it follows from (C) (K K) ∆ = Aα : α NN . 1 [16, Theorem× 1.2] that\ X has no free{ sequences∈ of length} ω1, i.e., t(X) ! ω. 2 2.13. Corollary. Under MA(ω ),S if X is a compact space such that X2 # is dominated by a Polish space then X is metrizable. 1 \ Theorem (Orihuela, Tkachuk, B.C. 2011) Proof. Apply Proposition 2.2 to see that the space X2 # is dominated by P so t(X) ω by Theorem 2.12 and hence X is \ ! metrizable by Theorem 2.9. (A) +2 (B) + (C) + MA(ω1) K is metrizable. In the rest of this section we study the spaces hereditarily dominated⇒ by a second countable space. The motivation here is a result of Hodel established in [14, Corollary 4.13]; it says that any hereditarily Lindelöf Σ-space is cosmic. We will look Proof.-at this hereditary property in function spaces to show that a somewhat stronger statement is true in a general situation under Martin’s Axiom. 1 The(A) following+ fact(B) is an+ immediate(C) + consequence MA(ω1 of) [26, PropositionK has 2.7]. small diagonal, i.e., for any uncountable set A (K K) ⇒∆ there exists an uncountable B A such 2.14. Proposition. If X is a space which has a countable network modulo a cover of X by countably compact sets then C p (X) is Lindelöf ⊂ × \ X ⊂ Σ-framed,that i.e.,B there is∆ a Lindelöf = /0.Σ-space L such that C p(X) L R . ∩ ⊂ ⊂ 2.15.2 Theorem.K hasA smallspace C p(X diagonal) is dominated by a secondK has countable countable space if and onlytightness if it is Lindelöf Σ. K K has ⇒ ⇒ × Proof.countableWe must only prove tightness; necessity. Suppose that C p(X) is dominated by a second countable space M and ﬁx a family F : K (M) which witnesses this. It follows from Proposition 2.14 and Proposition 2.6 that C (C (X)) is Lindelöf Σ- { K ∈ K } p p framed. Applying [21, Theorem 3.5] we conclude that υ(C p (X)) is a Lindelöf Σ-space and hence υ X is a Lindelöf Σ-space by [21, Corollary 3.6]. 208 B. Cascales et al. / Topology and its Applications 158 (2011) 204–214

2.11. Theorem. If X is a dyadic compact space and X 2 # is dominated by a second countable space then X is metrizable. \

Proof. If X is first countable then it is metrizable by [11, 3.12.12(e)]. Therefore we can assume that there exists a point x X of uncountable character in X.Apply[11,3.12.12(i)]tofindanuncountableone-pointcompactificationA of a discrete ∈ space such that A X and x is the unique non-isolated point of A.ThenB (A x ) x is an uncountable closed discrete ⊂ = \{ } × { } subspace of (X X) # while we have ext(X2 #) ω by Theorem 2.1(h), a contradiction. × \ \ = ψ(α) ψ(α), (closure in K K) 2 The above results show that, to prove that any⊂ compact space X with ×X2 # dominated by a second countable space is take x ψ(α); \ metrizable, it∈ suffices to show that any such space has a countable tightness. While we don’t know whether this implication is truethere in general, is A weψ( doα) present countable some with partialx progressA; in this direction. ⊂ ∈ is x A x ψ(α); 2 2.12. Theorem. Assume MA(ω1) and suppose that X is a compact space such that X # is P-dominated. Then X has a small diagonal ∈ ⇒ ∈ \ and henceotherwise, t(X) ωx. (A A) x is cluster point of a sequence in ψ(α) x ψ(α). = ∈ \ ⇒ ⇒ ∈ 2 Proof. Suppose that A zα: α < ω1 X # and α β implies zα zβ .FixaP-directed cover K p: p P of compact Proposition, B.2 C., 1987={ } ⊂ \ $= $= { ∈ } subsets of X #.Takepα P such that zα K pα for any α < ω1. \ ∈ ∈ It follows from MA( ) that there exists p such that p p for any < .ThesetP K : q and q p ω1N P N α !∗(K K) ∆ α ω1 q P ∗ Given X and Aα : α N as above, if we define∈ ψ : N 2 × \ given by = { ∈ = } is σ -compact and A P .Consequently,thereisq P for which Kq A is uncountable; therefore the set Kq A witnesses { ⊂∈ } ∈ → ∩ ! ∩ the small diagonal property of X.Sincenospacewithasmalldiagonalcanhaveaconvergent∞ ω1-sequence, it follows from [16, Theorem 1.2] that X has no free sequences of length ω , i.e., t(X) ω. ψ(α) := 1 Aβ : β m!= α m { | | 2} m\=1 [ 2.13. Corollary. Under MA(ω ), if X is a compact space such that X2 # is dominated by a Polish space then X is metrizable. then: 1 \ Proof. Apply Proposition 2.2 to see that the space X2 # is dominated by P so t(X) ω by Theorem 2.12 and hence X is each ψ(α) is countably compact (even more,\ all cluster points of any sequence! in ψ(α) remains in ψ(α)). metrizable by Theorem 2.9. if ψ(α) is compact then α ψ(α) gives K-analytic structure to (K K) ∆. 2 → × \ In the rest of this section we study the spaces hereditarily dominated by a second countable space. The motivation here is a result of Hodel established in [14, Corollary 4.13]; it says that any hereditarily Lindelöf Σ-space is cosmic. We will look at this hereditary property in function spaces to show that a somewhat stronger statement is true in a general situation under Martin’s Axiom. The following fact is an immediate consequence of [26, Proposition 2.7].

2.15. Theorem. A space C p(X) is dominated by a second countable space if and only if it is Lindelöf Σ.

Proof. We must only prove necessity. Suppose that C p(X) is dominated by a second countable space M and ﬁx a family F : K (M) which witnesses this. It follows from Proposition 2.14 and Proposition 2.6 that C (C (X)) is Lindelöf Σ- { K ∈ K } p p framed. Applying [21, Theorem 3.5] we conclude that (υK(C p (X))Kis) a Lindelöf∆ isΣ Lindel¨of-space and hence ∆υ X is aG Lindelöfδ ΣK-spaceis by [21,metrizable. Corollary 3.6]. ⇒ × \ ⇒ ⇒ ♣

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Metrizability of compact sets (II)

K compact space & Aα : α NN subsets of (K K) ∆. We write: { ∈ } × \ (A) each Aα is compact;

(B) Aα A whenever α β; ⊂ β ≤ (C) (K K) ∆ = Aα : α NN . × \ { ∈ } S Theorem (Orihuela, Tkachuk, B.C. 2011)

(A) + (B) + (C) + MA(ω1) K is metrizable. ⇒ Proof.-

1 (A) + (B) + (C) + MA(ω1) K has small diagonal, i.e., for any uncountable set A (K K) ⇒∆ there exists an uncountable B A such ⊂ × \ ⊂ that B ∆ = /0. ∩ 2 K has small diagonal K has countable tightness K K has countable tightness; ⇒ ⇒ × 3 (K K) ∆ is K-analytic × \ 208 B. Cascales et al. / Topology and its Applications 158 (2011) 204–214

2.11. Theorem. If X is a dyadic compact space and X 2 # is dominated by a second countable space then X is metrizable. \

2.15. Theorem. A space C p(X) is dominated by a second countable space if and only if it is Lindelöf Σ.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Metrizability of compact sets (II)

K compact space & Aα : α NN subsets of (K K) ∆. We write: { ∈ } × \ (A) each Aα is compact; Proposition, B. C., 1987

(B) Aα Aβ whenever α β; (K K) ∆ Given X and Aα : α NN as above, if we deﬁne ψ : NN 2 given by ⊂{ ∈ } ≤ → × \ (C) (K K) ∆ = Aα : α NN∞ . ψ(α) := A : β m = α m × \ { ∈ } { β | | } S m\=1 [ then: Theorem (Orihuela, Tkachuk, B.C. 2011) each ψ(α) is countably compact (even more, all cluster points of any sequence in ψ(α) remains in ψ(α)). if ψ(α) is compact(A) then+ (B)α ψ+(α)(C) gives+K-analyticMA(ω structure1) K to (K is metrizable.K) ∆. → ⇒ × \ Proof.-

2.11. Theorem. If X is a dyadic compact space and X 2 # is dominated by a second countable space then X is metrizable. \

2 2.12. Theorem. Assume MA(ω1) and suppose that X is a compact space such that X # is P-dominated. Then X has a small diagonal and hence t(X) ω. \ =

2 Proof. Suppose that A zα: α < ω1 X # and α β implies zα zβ .FixaP-directed cover K p: p P of compact 2 ={ } ⊂ \ $= $= { ∈ } subsets of X #.Takepα P such that zα K pα for any α < ω1. \ ∈ ∈ It follows from MA(ω1) that there exists p P such that pα ∗ p for any α < ω1.ThesetP Kq: q P and q ∗ p ∈ ! = { ∈ = } is σ -compact and A P .Consequently,thereisq P for which Kq A is uncountable; therefore the set Kq A witnesses ⊂ ∈ ∩ ! ∩ the small diagonal property of X.Sincenospacewithasmalldiagonalcanhaveaconvergentω1-sequence, it follows from [16, Theorem 1.2] that X has no free sequences of length ω1, i.e., t(X) ! ω. 2 2.13. Corollary. Under MA(ω ), if X is a compact space such that X2 # is dominated by a Polish space then X is metrizable. 1 \ Proof. Apply Proposition 2.2 to see that the space X2 # is dominated by P so t(X) ω by Theorem 2.12 and hence X is \ ! metrizable by Theorem 2.9. 2 In the rest of this section we study the spaces hereditarily dominated by a second countable space. The motivation here is a result of Hodel established in [14, Corollary 4.13]; it says that any hereditarily Lindelöf Σ-space is cosmic. We will look at this hereditary property in function spaces to show that a somewhat stronger statement is true in a general situation under Martin’s Axiom. The following fact is an immediate consequence of [26, Proposition 2.7].

2.15. Theorem. A space C p(X) is dominated by a second countable space if and only if it is Lindelöf Σ.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions

ψ(α) ψ(α), (closure in K K) ⊂ × Metrizabilitytake x ψ(α); of compact sets (II) ∈ there is A ψ(α) countable with x A; ⊂ ∈ is x A x ψ(α); K compact∈ ⇒ space∈ & A : α N subsets of (K K) ∆. We write: otherwise, x (A A) xαis clusterN point of a sequence in ψ(α) x ψ(α). ∈ \ {⇒ ∈ } ×⇒ ∈\ (A) each Aα is compact; Proposition, B. C., 1987

2.11. Theorem. If X is a dyadic compact space and X 2 # is dominated by a second countable space then X is metrizable. \

Proof. If X is first countable then it is metrizable by [11, 3.12.12(e)]. Therefore we can assume that there exists a point x X of uncountable character in X.Apply[11,3.12.12(i)]tofindanuncountableone-pointcompactificationA of a discrete ∈ space such that A X and x is the unique non-isolated point of A.ThenB (A x ) x is an uncountable closed discrete ⊂ = \{ } × { } subspace of (X X) # while we have ext(X2 #) ω by Theorem 2.1(h), a contradiction. × \ \ = ψ(α) ψ(α), (closure in K K) 2 The above results show that, to prove that any⊂ compact space X with ×X2 # dominated by a second countable space is take x ψ(α); \ metrizable, it∈ suffices to show that any such space has a countable tightness. While we don’t know whether this implication is truethere in general, is A weψ( doα) present countable some with partialx progressA; in this direction. ⊂ ∈ is x A x ψ(α); 2 2.12. Theorem. Assume MA(ω1) and suppose that X is a compact space such that X # is P-dominated. Then X has a small diagonal ∈ ⇒ ∈ \ and henceotherwise, t(X) ωx. (A A) x is cluster point of a sequence in ψ(α) x ψ(α). = ∈ \ ⇒ ⇒ ∈ 2 Proof. Suppose that A zα: α < ω1 X # and α β implies zα zβ .FixaP-directed cover K p: p P of compact Proposition, B.2 C., 1987={ } ⊂ \ $= $= { ∈ } subsets of X #.Takepα P such that zα K pα for any α < ω1. \ ∈ ∈ It follows from MA( ) that there exists p such that p p for any < .ThesetP K : q and q p ω1N P N α !∗(K K) ∆ α ω1 q P ∗ Given X and Aα : α N as above, if we define∈ ψ : N 2 × \ given by = { ∈ = } is σ -compact and A P .Consequently,thereisq P for which Kq A is uncountable; therefore the set Kq A witnesses { ⊂∈ } ∈ → ∩ ! ∩ the small diagonal property of X.Sincenospacewithasmalldiagonalcanhaveaconvergent∞ ω1-sequence, it follows from [16, Theorem 1.2] that X has no free sequences of length ω , i.e., t(X) ω. ψ(α) := 1 Aβ : β m!= α m { | | 2} m\=1 [ 2.13. Corollary. Under MA(ω ), if X is a compact space such that X2 # is dominated by a Polish space then X is metrizable. then: 1 \ Proof. Apply Proposition 2.2 to see that the space X2 # is dominated by P so t(X) ω by Theorem 2.12 and hence X is each ψ(α) is countably compact (even more,\ all cluster points of any sequence! in ψ(α) remains in ψ(α)). metrizable by Theorem 2.9. if ψ(α) is compact then α ψ(α) gives K-analytic structure to (K K) ∆. 2 → × \ In the rest of this section we study the spaces hereditarily dominated by a second countable space. The motivation here is a result of Hodel established in [14, Corollary 4.13]; it says that any hereditarily Lindelöf Σ-space is cosmic. We will look at this hereditary property in function spaces to show that a somewhat stronger statement is true in a general situation under Martin’s Axiom. The following fact is an immediate consequence of [26, Proposition 2.7].

2.15. Theorem. A space C p(X) is dominated by a second countable space if and only if it is Lindelöf Σ.

Proof. We must only prove necessity. Suppose that C p(X) is dominated by a second countable space M and ﬁx a family F : K (M) which witnesses this. It follows from Proposition 2.14 and Proposition 2.6 that C (C (X)) is Lindelöf Σ- { K ∈ K } p p framed. Applying [21, Theorem 3.5] we conclude that υ(C p (X)) is a Lindelöf Σ-space and hence υ X is a Lindelöf Σ-space by [21, Corollary 3.6].

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Metrizability of compact sets (II)

K compact space & Aα : α NN subsets of (K K) ∆. We write: { ∈ } × \ (A) each Aα is compact;

(B) Aα A whenever α β; ⊂ β ≤ (C) (K K) ∆ = Aα : α NN . × \ { ∈ } S Theorem (Orihuela, Tkachuk, B.C. 2011)

(A) + (B) + (C) + MA(ω1) K is metrizable. ⇒ Proof.-

3 (K K) ∆ is K-analytic (K K) ∆ is Lindel¨of ∆ is Gδ K is metrizable.× \ ⇒ × \ ⇒ ⇒ ♣ Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions

Domination by Second Countable Spaces Theorem (Orihuela, Tkachuk, B.C. 2011) For a topological space TFAE: 1 X is countably K-determined; 2 X is Dieudonn´ecomplete and dominated by a second countable space.

The class of spaces dominated by a second countable space enjoy the usual stability properties we might expect.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Domination by Second Countable Spaces

Deﬁnition A topological space X is dominated by a second countable space, if there is a second countable space M and a family AK : K K (M) X such that: { ∈ } ⊂ (A) each AK is compact;

(B) AK AF whenever K F ; ⊂ ⊂ (C) X = AK : K K (M) . { ∈ } S The class of spaces dominated by a second countable space enjoy the usual stability properties we might expect.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Domination by Second Countable Spaces

Deﬁnition A topological space X is dominated by a second countable space, if there is a second countable space M and a family AK : K K (M) X such that: { ∈ } ⊂ (A) each AK is compact;

(B) AK AF whenever K F ; ⊂ ⊂ (C) X = AK : K K (M) . { ∈ } S Theorem (Orihuela, Tkachuk, B.C. 2011) For a topological space TFAE: 1 X is countably K-determined; 2 X is Dieudonn´ecomplete and dominated by a second countable space. Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Domination by Second Countable Spaces

Deﬁnition A topological space X is dominated by a second countable space, if there is a second countable space M and a family AK : K K (M) X such that: { ∈ } ⊂ (A) each AK is compact;

The class of spaces dominated by a second countable space enjoy the usual stability properties we might expect. If X is dominated by a second countable space, if there is a second countable space M and a family A : K K (M) such that: { K ∈ } (A) each AK is compact; (B) A A whenever K F ; K ⊂ F ⊂ (C) X = A : K K (M) . { K ∈ } We take: T :=S (K (M),h), ϕ(K) := AK and we can generate the USCO ψ in many cases.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Techniques

Generation of usco maps, Orihuela and B. C. 1991 Let T be a ﬁrst-countable, X a topological space and let ϕ : T 2X be a set-valued map satisfying the property →

ϕ(tn) is relatively compact for each convergent sequence (tn)n in T . (1) n N [∈ If for each x in X we deﬁne

C(t) := x X : there is tn t in T , for every n N there is { ∈ → ∈ xn ϕ(tn) and x is cluster point of (xn)n . ∈ } Then: each C(t) is countably compact. if ψ(t) := C(t) is compact then t ψ(t) is usco ψ : T K (X ). → → We take: T := (K (M),h), ϕ(K) := AK and we can generate the USCO ψ in many cases.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Techniques

Generation of usco maps, Orihuela and B. C. 1991 Let T be a ﬁrst-countable, X a topological space and let ϕ : T 2X be a set-valued map satisfying the property →

ϕ(tn) is relatively compact for each convergent sequence (tn)n in T . (1) n N [∈ IfIfX foris dominated each x byin a secondX we countable deﬁne space, if there is a second countable space M and a family A : K K (M) such that: { K ∈ } (A) each AK is compact; C(t) := x X : there is tn t in T , for every n N there is (B) A A whenever K F ; K ⊂ {F ∈ ⊂ → ∈ (C) X = A : K K (M) . xn ϕ(tn) and x is cluster point of (xn)n . { K ∈ } ∈ } S Then: each C(t) is countably compact. if ψ(t) := C(t) is compact then t ψ(t) is usco ψ : T K (X ). → → Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Techniques

Generation of usco maps, Orihuela and B. C. 1991 Let T be a ﬁrst-countable, X a topological space and let ϕ : T 2X be a set-valued map satisfying the property →

ϕ(tn) is relatively compact for each convergent sequence (tn)n in T . (1) n N [∈ IfIfX foris dominated each x byin a secondX we countable deﬁne space, if there is a second countable space M and a family A : K K (M) such that: { K ∈ } (A) each AK is compact; C(t) := x X : there is tn t in T , for every n N there is (B) A A whenever K F ; K ⊂ {F ∈ ⊂ → ∈ (C) X = A : K K (M) . xn ϕ(tn) and x is cluster point of (xn)n . { K ∈ } ∈ } S Then:We take: T := (K (M),h), ϕ(K) := AK and we can generate the USCO ψ in many cases. each C(t) is countably compact. if ψ(t) := C(t) is compact then t ψ(t) is usco ψ : T K (X ). → → Theorem (Orihuela, Tkachuk, B.C. 2011) Let K be a compact space. If there is a second countable space M and a family AF : F K (M) (K K) ∆ such that: { ∈ } ⊂ × \ (A) each AF is compact;

(B) AF AL whenever F L; ⊂ ⊂ (C) (K K) ∆ = AF : F K (M) . × \ { ∈ } and S

(D) every compact subset of (K K) ∆ is contained in some AF . × \ Then K is metrizable.

Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Two noticeable results

Theorem (Orihuela, Tkachuk, B.C. 2011)

Cp(X ) is countably K-determined iﬀ is dominated by a second countable space. Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Two noticeable results

Theorem (Orihuela, Tkachuk, B.C. 2011)

Cp(X ) is countably K-determined iﬀ is dominated by a second countable space.

Theorem (Orihuela, Tkachuk, B.C. 2011) Let K be a compact space. If there is a second countable space M and a family AF : F K (M) (K K) ∆ such that: { ∈ } ⊂ × \ (A) each AF is compact;

(B) AF AL whenever F L; ⊂ ⊂ (C) (K K) ∆ = AF : F K (M) . × \ { ∈ } and S

(D) every compact subset of (K K) ∆ is contained in some AF . × \ Then K is metrizable. Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions Ready to ﬁnish... I kept the promise

We know have to solve the exercises and a bit more that become a Theorem For a compact space K TFAE: a. . . 1 K is metrizable;

2 (C(K), ∞) is separable; k · k 3 ∆ is a Gδ ; 4 ∆ = nGn with Gn open and Gn n a basis of neighb. of ∆; ∩ { } 5 (K K) ∆ = nFn, with Fn an increasing fundamental × \ ∪ { } family of compact sets in (K K) ∆; × \ N 6 (K K) ∆ = Aα : α N with each Aα a fundamental × \ { ∈ } { } family of compact sets such that Aα A whenever α β; S ⊂ β ≤ 7 (K K) ∆ is Lindel¨of. × \ Proto Idea Starting Point Domination by Polish&Appl. Domination by second countable Extensions&Open questions

Open questions Beware!!!

1 This lecture and the paper DO ONLY SHARE the results. 2 None of the proofs presented here are in the paper.