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Fans and Their Applications in General Topology, Functional Analysis And Fans and their applications in General Topology, Functional Analysis and Topological Algebra Taras Banakh Ivan Franko National University of Lviv (Ukraine) Jan Kochanowski University in Kielce (Poland) E-mail address: [email protected] arXiv:1602.04857v2 [math.GN] 17 Feb 2016 2010 Mathematics Subject Classification. Primary 54D50; 54E20; 54H10; 46B20; Secondary 08B20; 22A05; 22A15; 22A30; 54B30; 54C35; 54D55; 54E18; 54G10; 54G12; 54H11 Key words and phrases. k-space, Ascoli space, fan, strong fan, function space, Banach space, weak topology, rectifiable space, functor, category, free topological algebra, free topological group. Contents Chapter 1. Introduction and survey of main results 6 Chapter 2. Some notation and general tools 14 2.1. Some standard notations 14 2.2. Almost disjoint families of sets 14 2.3. Functionally Hausdorff spaces 14 2.4. Bounded sets in topological spaces 15 2.5. kω-Spaces 15 2.6. k-Spaces, k-homeomorphisms and k-metrizable spaces 16 2.7. Hemicompact spaces and k-sums of hemicompact spaces 17 2.8. Compact-finite sets in topological spaces 17 2.9. (Strongly) compact-finite families in ℵ-spaces 18 2.10. Extension operators and Ck-embedded subsets 19 Chapter 3. Applications of strict Cld-fans in General Topology 21 3.1. Characterizing P -spaces and discrete spaces 21 3.2. Characterizing kR-spaces and scattered sequential spaces 22 3.3. Two characterizations of spaces containing no strict Cld-fans 23 3.4. Preservation of (strong) fans by some operations 24 3.5. Some counterexamples 26 Chapter 4. S-Fans, S-semifans and D-cofans in topological spaces 28 4.1. S-fans 28 4.2. S-semifans 30 4.3. Dω-Cofans 33 4.4. (Strong) Fin-fans in products 36 4.5. Constructing (strong) Finω-fans in rectifiable spaces 38 Chapter 5. Strong Fin-fans in functions spaces 40 ω 5.1. A duality between Dω-cofans and S -fans in function spaces 40 5.2. Function spaces Ck(X) 42 5.3. Function spaces Ck(X, 2) 43 Chapter 6. Strong Finω-fans in locally convex and Banach spaces with the weak topology 46 6.1. Strong Finω-fans in locally convex spaces with the weak topology 46 6.2. Strong Finω-fans in bounded subsets of Banach spaces with the weak topology 46 Chapter 7. Topological functors and their properties 49 7.1. Functors with finite supports 49 7.2. I-regular functors 52 7.3. Bounded and strongly bounded functors 53 7.4. HM-commuting functors 54 7.5. Functors preserving closed embeddings of metrizable compacta 56 7.6. Functors preserving preimages 57 7.7. Strong Cld-fans in functor-spaces 59 Chapter 8. Constructing Fin-fans in topological algebras 61 8.1. Topological algebras of type x·y 61 8.2. Topological algebras of type x(y∗z) 63 3 4 CONTENTS 8.3. Topological algebras of type (x∗y)(z∗s) 65 ∗ <ω 8.4. Topological algebras of type x(s yi) 67 8.5. Topological algebras of type x(y(s∗z)y∗) 69 8.6. Topological algebras of type (x(s∗z)x∗)·(y(s∗z)y∗) 70 8.7. Algebraic structures on monadic functors 71 8.8. Superbounded functors 76 Chapter 9. Constructing fans in free universal algebras 79 9.1. Universal E-algebras 79 9.2. Topologized and topological E-algebras 79 9.3. Free topologized E-algebras in HM-stable varieties 82 9.4. Free topologized E-algebras in d+HM-stable varieties 83 9.5. Free topological E-algebras in kω-stable varieties 84 9.6. Free topologized E-algebras in d+HM+kω-stable varieties 85 9.7. Paratopological (E, C)-algebras 87 9.8. Fans in free paratopological E˙ -algebras 89 9.9. Fans in free paratopological E˙ ∗-algebras 91 ∗ 9.10. Free paratopological E˙ -algebras in d+HM+kω-superstable varieties 93 Chapter 10. Free objects in some varieties of topologized algebras 96 10.1. Magmas and topologized magmas 96 10.2. Topologized ∗-magmas 97 10.3. Free topological semigroups 98 10.4. Free topological semilattices 99 10.5. Free Lawson semilattices 100 10.6. Free topological inverse semigroups 102 10.7. Free topological Abelian groups 102 10.8. Free topological groups 104 10.9. Free paratopological Abelian groups 105 10.10. Free paratopological groups 105 10.11. Free locally convex spaces 106 10.12. Free linear topological spaces 110 10.13. Summary 112 Bibliography 113 Index 115 CONTENTS 5 Abstract. A family (Fα)α∈λ of closed subsets of a topological space X is called a (strict) Cld-fan in X if this family is (strictly) compact-finite but not locally finite in X. Applications of (strict) Cld-fans are based on a simple observation that k-spaces contain no Cld-fan and Ascoli spaces contain no strict Cld-fan. In this paper we develop the machinery of (strict) fans and apply it to detecting the k-space and Ascoli properties in spaces that naturally appear in General Topology, Functional Analysis, and Topological Algebra. In particular, we detect (generalized) metric spaces X whose functor-spaces, functions spaces, free (para)topological (abelian) groups, free (locally convex) linear topological spaces, free (Lawson) topological semilattices, and free (para)topological (Clifford, Abelian) inverse semigroups are k-spaces or Ascoli spaces. CHAPTER 1 Introduction and survey of main results In this paper we isolate one property of topological spaces (namely, the presence of a strong Fin-fan) which is responsible for the failure of the k-space (more precisely, the Ascoli) property in many natural spaces such as (free) topological groups, function spaces, or Banach spaces with the weak topology. Let us recall that a topological space X is a k-space if each k-closed subset of X is closed. A subset A ⊂ X is called k-closed if for every compact subset K ⊂ X the intersection A ∩ K is closed in K. Definition 1.1. Let λ be a cardinal. An indexed family (Xα)α∈λ of subsets of a topological space X is called • locally finite if any point x ∈ X has a neighborhood Ox ⊂ X such that the set {α ∈ λ : Ox ∩ Xα 6= ∅} is finite; • compact-finite (resp. compact-countable) in X if for each compact subset K ⊂ X the set {α ∈ λ : K ∩ Xα 6= ∅} is finite (resp. at most countable); • strongly compact-finite in X if each set Xα has an R-open neighborhood Uα ⊂ X such that the family (Uα)α∈A is compact-finite in X; • strictly compact-finite in X if each set Xα has a functional neighborhood Uα ⊂ X such that the family (Uα)α∈A is compact-finite in X; Now we explain some undefined notions appearing in this definition. A subset U of a topological space X is called R-open if for each point x ∈ U there is a continuous function f : X → [0, 1] such that f(x) = 1 and f(X \ U) ⊂{0}. It is clear that each R-open set is open. The converse is true for open subsets of Tychonoff spaces. A subset U of a topological space X is called a functional neighborhood of a set A ⊂ X if there is a continuous function f : X → [0, 1] such that f(A) ⊂{0} and f(X \ U) ⊂{1}. It is clear that each functional neighborhood U of a set A in a topological space X contains an R-open neighborhood of the closure A¯ of A in X. The converse is true if A¯ is compact. In a normal space X each neighborhood of a closed subset A ⊂ X is functional. Now we are able to introduce the notion of a fan, the principal notion of this paper. Definition 1.2. Let X be a topological space and λ be a cardinal. An indexed family (Fα)α∈λ of subsets of X is called a fan (more precisely, a λ-fan) in X if this family is compact-finite but not locally finite in X. A fan (Fα)α∈λ is called strong (resp. strict) if each set Fα has a (functional) R-open neighborhood Uα ⊂ X such that the family (Uα)α∈λ is compact-finite. Since each strictly compact-finite family is strongly compact-finite, we get the following implications: strict fan ⇒ strong fan ⇒ fan. If all sets Fα of a λ-fan (Fα)α∈λ belong to some fixed family F of subsets of X, then the fan will be called an F-fan (more precisely, an F λ-fan) in X. Two instances of the family F will be especially important for our purposes: the family Fin of finite subsets of X and the family Cld of closed subsets of X. In these two cases F-fans will be called Fin-fans and Cld-fans, respectively. Since each finite subset of a T1-space is closed, each Fin-fan in a T1-space is a Cld-fan. Taking into account that each R-open neighborhood of a finite subset F ⊂ X is a functional neighborhood of F in X, we conclude that an Fin-fan is strong if and only it is strict. So, in T1-spaces we get the implications: strict Fin-fan ks +3 strong Fin-fan +3 Fin-fan strict Cld-fan +3 strong Cld-fan +3 Cld-fan. Applications of Cld-fans to k-spaces are based on the following simple observation. 6 1. INTRODUCTION AND SURVEY OF MAIN RESULTS 7 Proposition 1.3. A k-space contains no Cld-fans and no Fin-fans. Proof. It suffices to check that each compact-finite family (Fα)α∈λ of closed subsets of a k-space X is locally finite at each point x ∈ X.
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