Analytic Baire Spaces

Analytic Baire spaces

A. J. Ostaszewski Mathematics Dept., London School of Economics London WC2A 2AE, UK E-mail: [email protected]

Abstract We generalize to the non-separable context a theorem of Levi characterizing Baire analytic spaces. This allows us to prove a joint- continuity result for non-separable normed groups, previously known only in the separable context.

1 Introduction

This paper is inspired by Sandro Levi’s[Levi], similarly titled ‘On Baire cosmic spaces’,containing an Open Mapping Theorem (a result of the Direct Baire Property given below) and a useful corollary on comparison of topologies; all these results are in the separable realm. Here we give non-separable generalizations (see Main Theorem 1.6) and, as an illustration of their use- fulness, Main Theorem 1.9 o¤ers a non-separable version of an Ellis-type theorem (see [Ell1, Cor. 2], cf. [Bou1], [Bou2], and the more recent [SolSri]) with a ‘one-sided’ continuity condition implying that a right-topological group generated by a right-invariant metric (i.e. a normed group in the terminology of §4) is a topological group. Unlike Ellis we do not assume that the group is abelian, nor that it is locally compact; the non-separable context requires some preservation of -discreteness as a side-condition (see below). Given that the application in mind is metrizable, references to non- separable descriptive theory remain, for transparency, almost exclusively in

2010 Mathematics Subject Classi…cation: Primary 26A03; 04A15; Secondary 02K20. Key words and phrases: automatic continuity, analytic sets, analytic Baire theorem, analytic Cantor theorem, Nikodym’sStability Theorem, non-separable descriptive topology, shift-compactness, non-commutative groups, group-norm, open mapping.

1 2 A. J. Ostaszewski the metric realm, though we do comment on the regular Hausdor¤ context in §5 (see Remark 5.4). Levi’swork draws together two notions: BP –the Baire set property (i.e. that a set is open modulo a meagre set, so ‘almost open’), and BS – the Baire space property (i.e. that Baire’s theorem holds in the space). Below we keep the distinction clear by using the terms ‘Baire property’and ‘Baire space’.The connection between BP and BS is not altogether surprising, and as we explain in Remark 4.5 the two are ‘almost’the same in a precise sense, at least in the context of normed groups (cf. [Ost-S], where this closeness is fully exploited). We refer to [Eng] for general topological usage (though we prefer ‘meagre’ as a term). We say that a subspace S of a metric space X has a Souslin- (X) representation if there is a ‘determining’system F (i n) := F (i n): F h j i h j i NN of sets in (X) (the closed sets) with 2 i F S = F (i n); where I = NN i I n N 2 2 j [ \ and i n denotes (i1; :::; in): We will say that a topological space is classi- j cally analytic if it is the continuous image of a Polish space (Levi terms these ‘Souslin’) and not necessarily metrizable, in distinction to an (absolutely) analytic space, i.e. one that here is metrizable and is embeddable as a Souslin- set in its own metric completion; in particular, in a complete F metric space -subsets (being ) are analytic. We call a Hausdor¤ space G F almost analytic if it is analytic modulo a meagre set. Similarly, a space X0 is absolutely , or an absolute- , if X0 is a in all spaces X containing X0 G G G as a subspace. (This is equivalent to complete metrizability in the narrowed realm of metrizable spaces [Eng, Th. 4.3.24], and to topological/Cechµ completeness in the narrowed realm of completely regular spaces –[Eng, §3.9].)

So a metrizable absolute- is analytic; we use this fact in Lemma 6.2. G Levi’sresults follow from the following routine observation.

Theorem 1.1 (On the Direct Baire Property, [Levi]). Let X be a classically analytic space and Y Hausdor¤. Every continuous map f : X ! Y has the direct Baire property: the image of any open set in X has the Baire property in Y:

The nub of the theorem is that, with X as above, continuity preserves various analyticity properties such as that open, and likewise closed, sets are taken to analytic sets, in brief: a continuous map is open-analytic and closed-analytic in the terminology of [Han-74], and so preserves the Baire Analytic Baire Spaces 3 property. (See below in Remarks 1.5.3 for a reprise of this theme.) Levi deduces the following characterization of Baire spaces in the category of classically analytic spaces. Theorem 1.2 (Levi’s Open Mapping Theorem, [Levi]). Let X be a regular classically analytic space. Then X is a Baire space i¤ X = f(P ) for some continuous map f on some Polish space P with the property that there exists a subspace X0 X which is a dense metrizable absolute- such that 1 G the restriction map f P 0 : P 0 X0 is open for P 0 = f (X0). j ! The result may be regarded as implying an “inner regularity”property (compare the capacitability property) of a classically analytic space X: if

X is a Baire space, then X contains a dense absolute- subspace, so a G Baire space. The existence of a dense completely metrizable subspace – making X almost complete in the sense of Frolík ([Frol], but the term is due to Michael [Mich91]) –is a result that implicitly goes back to Kuratowski ([Kur-1] IV.2 p. 88, because a classically analytic set has the Baire property in the restricted sense – Cor. 1 p. 482). Generalizations of the latter result, including the existence of a restriction map that is a homeomorphism between a -subset and a dense set, is given by Michael in [Mich86]; but G there the continuous map f requires stronger additional properties such as openness on P (unless P is separable), which Levi’sresult delivers. Theorem 1.2 has a natural extension characterizing a Baire space (in the same way) when it is almost analytic. Indeed, with X0 as above, the space X is almost complete and so almost analytic. On the other hand, if X is a Baire space and almost analytic, then by supressing a meagre and passing to F an absolutely -subspace, we may assume that X is a Baire space which G is analytic, so has the open mapping representation of the theorem, and in particular is almost complete (for more background see [Ost-S]; cf. Cor. 1.8). Since an analytic space is a continuous image, Theorem 1.2 may be viewed as an ‘almost preservation’result for complete metrizability under continuity in the spirit of the classical theorem of Hausdor¤ (resp. Vain- stein) on the preservation of complete metrizability by open (resp. closed) continuous mappings –see Remarks 1.11.4 at the end of this section for the most recent improvements and the literature of preservation. We note that Michael [Mich91, Prop. 6.5] shows that almost completeness is preserved by demi-open maps (i.e. continuous maps under which inverse images of dense open sets are dense). Theorem 1.2 has an interesting corollary on the comparison of re…nement topologies. For a discussion of re…nements see [Ost-S, 4 A. J. Ostaszewski

§7.1] (for examples of completely metrizable and of analytic re…nements see [Kech, Th. 13.6, Th. 25.18, Th. 25.19]).

Theorem 1.3 (Levi’s Comparison Theorem, [Levi]). For ; 0 two T T topologies on a set X with (X; 0) classically analytic (e.g. Polish) and 0 T T re…ning (i.e. 0), if (X; ) is a regular Baire space, then there is a T T T T -dense ( ) -set on which and 0 agree. T G T T T We o¤er a generalization in Main Theorem 1.6 below to the broader category of (absolutely) analytic spaces –be they separable or non-separable metric spaces. We will need the following de…nitions (see below for com- ments). Recall that a Hausdor¤ space X is paracompact ([Eng], Ch. 5) if every open cover of X has a locally-…nite open re…nement, and further that for an (indexed) family := Bt : t T : B f 2 g (i) is index-discrete in the space X (or just discrete when the index set B T is understood) if every point in X has a neighbourhood meeting the sets

Bt for at most one t T; 2 (ii) is -discrete (abbreviated to -d) if = n where each set n is B B n B B discrete as in (i), and S (iii) is a base for if every member of may be expressed as the union of B E E a subfamily of . For a topology (the family of all open sets) with B T B T a base for , this reduces to being simply a (topological) base. T B De…nitions 1.4 ([Han-74], §3; cf. [Han-71] §3.1 and [Mich82] Def. 3.3). 1. Call f : X Y base--discrete (or co--discrete) if the image under ! f of any discrete family in X has a -discrete base in Y . We need two re…nements that are more useful and arise in practice: call f : X Y an ! analytic (resp. Baire) base--discrete map (henceforth A--d , resp. B--d map) if in addition, for any discrete family of analytic sets in X; the E family f( ) has a -d base consisting of analytic sets (resp. sets with the E Baire property) in Y . We explain in §2 (Th. 2.6 and thereafter) why A--d maps, though not previously isolated, are really the only base--discrete maps needed in practice in analytic space theory.

2. ([Han-74], §2) An indexed family := At : t T is -discretely A f 2 g decomposable (-d decomposable) if there are discrete families n := Atn : A f t T such that At = Atn for each t: (The open family ( r; r): r R 2 g n f 2 g on the real line has a -d base, but is not -d decomposable –see [Hans-73b, S §3].) 3. ([Mich82] Def. 3.3) Call f : X Y index--discrete if the image under ! Analytic Baire Spaces 5 f of any discrete family in X is -d decomposable in Y: (Note f( ) is E E regarded as indexed by , so could be discrete without being index-discrete; E this explains the pre…x ‘index-’in the terminology here.) An index--discrete function is A--d (analytic base--discrete): see Th. 2.6 below. Remarks 1.5. Recall Bing’s Theorem ([Eng, Th. 4.4.8]) that a regular space is metrizable i¤ it has a -discrete base. In a separable space discrete sets are at most countable. So all the notions above generalize various as- pects of countability; in particular, in a separable metric setting all maps are (Baire) base--discrete. We comment brie‡y on their standing. (The paper [Han-74] is the primary source for these.) 1. In (3) above f has a stronger property than base--discreteness. For a proof see [Han-74] Prop. 3.7 (i); cf. [Mich82, Prop 4.3] shows that f with closed …bres has (3) i¤ it is base--discrete and has …bres that are 1- @ compact, i.e. separable (in the metric setting). The stronger property is often easier to work with than Baire base--discreteness; in any case the concepts are close, since for metric spaces and an in…nite cardinal: X is a base--discrete continuous image of N i¤ X is an index--discrete continuous image of a closed subset of N; both equivalent to analyticity ([Han-74] Th. 4.1), cf. Prop. 2.9 below. (See [Han-92] Th. 4.2, or [Han-74] Th. 4.1. In fact the natural continuous index--discrete representation of an analytic set has separable …bres, for which see §2 below; for a study of …bre condi- tions see [Han-98].) 2. Base--discrete continuous maps (in particular, Baire base--discrete and index--discrete continuous maps) preserve analyticity ([Han-74] Cor. 4.2). 3. If X is metric and (absolutely) analytic and f : X Y is injective and ! closed-analytic (or open-analytic), then f is base--discrete ([Han-74] Prop. 3.14). Base--discreteness is key to this paper just as open-analyticity is key to the separable context of the Levi results above. 4. If = ; with each discrete, is a -d base for the metrizable n N n n B 2 B B space X and each f( n) is -d decomposable, then f is index--discrete, S B and so base--discrete ([Han-74] Prop. 3.9).

5. A discrete collection = At : t T comprising analytic sets has the A f 2 g property that any subfamily has analytic union, i.e. is ‘completely additive analytic’.It turns out that in an analytic space a disjoint (or a point-…nite) collection is completely additive analytic i¤ it is -d decomposable (see A [KP] generalizing the disjoint case in [Han-71] Th. 2; see also [Fh]). By the proof of Th. 2.6 the decompositions can be into analytic sets. Main Theorem 1.6 (Generalized Levi Open Mapping Theorem – 6 A. J. Ostaszewski

Non-separable case). Let X be an analytic space (more generally, paracompact and -analytic – as de…ned in §2). Then X is a Baire space i¤ K X = f(P ) for some continuous, index--discrete map f on a completely metrizable space P with the property that there exists a dense completely metrizable subset X0 of X such that the restriction f P 0 : P 0 X0 is an G 1 j ! open mapping, with P 0 = f (X0); again a topologically complete subspace.

This is proved in §3. For related results on restriction maps of other special maps, see Michael [Mich91] §7. (Compare also [Han-92], Th. 6.4 and 6.25.) As immediate corollaries one has:

Corollary 1.7 (Generalized Levi Comparison Theorem). If ; 0 T T are two topologies on a set X with (X; ) a regular Baire space, and 0 T T an absolutely analytic (e.g. completely metrizable) re…nement of such T that every 0-index-discrete collection is -discretely decomposable under T , then there is a -dense ( ) -set on which and 0 agree. T T G T T T

Proof. For f take the identity map from (X; 0) to (X; ); which is T T continuous and index--discrete.

Corollary 1.8 (Almost Completeness Theorem). X is almost analytic and a Baire space i¤ X is almost complete.

Proof. If X contains a co-meagre analytic subspace A, then by Th. 1.6

A, being a Baire space, contains a dense completely metrizable subspace X0 which is co-meagre in X. So X is almost complete.

For a sharper characterization in the case of normed groups see [Ost-LB3, Th. 2], cf. Th. 4.4.. Corollary 1.7 will enable us to prove (in §6) the automatic continuity result of Main Theorem 1.9 below for right-topological groups with a right-invariant metric dR (the normed groups of §4). We write 1 1 dL(x; y) := dR(x ; y ); which is left-invariant, and dS := max dR; dL for f g the symmetrized (‘ambidextrous’)metric. The basic open sets under dS take R L the form B (x) B (x); i.e. an intersection of balls of "-radius under dR " \ " and dL centered at x; giving the join (coarsest common re…nement) of dR and dL: Following [Ost-Joint], for P a topological property it is convenient to say that the metric space (X; dR) is topologically symmetrized-P, or just semi-P, if (X; dS) has property P: In particular (X; dR) is semi-complete if

(X; dS) is topologically complete. As dR and dL are isometric under inversion, (X; dR) is semi-complete i¤ (X; dL) is. Analytic Baire Spaces 7

Main Theorem 1.9 (Semi-Completeness Theorem, cf. [Ost-Joint]).

For a normed group X, if (X; dR) is semi-complete and a Baire space, and the continuous embedding map j :(X; dS) (X; dR) is Baire base- ! -discrete (e.g. index--discrete), in particular this is so if X is separable, then the right and left uniformities of dR and dL coincide and so (X; dR) is a topologically complete topological group.

The following result will be needed later in conjunction with Theorem 5.2.

Lemma 1.10. For a normed group X, if the continuous embedding map j :(X; dS) (X; dR) is index--discrete (resp. base--discrete), then so ! 1 also is the inversion mapping from (X; dR) to (X; dR), i.e. i : x x : !

Proof. Suppose is a family of sets that is discrete in (X; dR); then 1 1 V := V : V is a family of sets that is discrete in (X; dL): As the V f 2 Vg 1 dS topology re…nes the dL topology, is discrete in (X; dS): Assuming V 1 1 that j is index--discrete (resp. base--discrete), the family j( ) = V V is -d decomposable (has a -dbase) in (X; dR): So inversion maps to a 1 V family that is -ddecomposable (has -dbase) in (X; dR): V Remarks 1.11. 1. Theorem 1.9 generalizes a classical result for abelian locally compact groups due to Ellis [Ell1]. 2. Remarks 1.5 noted that the ‘index--discreteness condition’imposed in the theorem is a natural one from the perspective of the non-separable theory of analytic sets, and the Lemma interprets this in terms of inversion, compare point 4 below. 3. For separable spaces, where discrete families are countable and so the embedding j above automatically preserves -discreteness, the result here was proved in [Ost-Joint] (to which we refer for the literature) in the form that a semi-Polish, normed group X; Baire in the right norm-topology, is a topologically-complete topological group. Rephrased in the language of uniformities generated by the norm ([Kel, Ch. 6 Pb. O]), this says that a normed group, Polish in the ambidextrous uniformity and Baire in either of the right or left uniformities, has coincident right and left uniformities, and so is a topological group. Key to its proof is that a continuous image of a complete separable metric space is a classically analytic space. So the ‘index -discreteness condition’is exactly the condition that secures preservation of analyticity. In the non-separable context continuity is not enough to preserve analyticity, and an additional property is needed, involving -d as above: see [Han-98] Example 4.2 for a non-analytic metric space that is 8 A. J. Ostaszewski a one-to-one continuous image of N for some uncountable cardinal (so a continuous image of a countable product of discrete, hence absolutely analytic, spaces of cardinal ). 4. Recent work by Holický and Pol [HP], in response to Ostrovsky’s in- sights and based on [Mich86, esp. §6] (which itself goes back to [GM]), connects preservation of (topological) completeness under continuous maps between metric spaces to the classical notion of resolvable sets. (The latter notion provides the natural generalization to Ostrovsky’s setting; see also Holický [H] for non-metrizable spaces.) They …nd that a map f preserves completeness if it ‘resolves countable discrete sets’,i.e. for every countable metrically-discrete set C and open neighbourhood V of C there is L with C L V such that f(L) is a resolvable set. Consider the implications for a group X with right-invariant metric dR (see above), when for f one takes j the identity embedding j :(X; dS) (X; dR); ! 1 and C = cn : n N is a dS-discrete set (so that C and C are f 2 g dR-discrete). To obtain the desired resolvability for j; it is necessary and su¢ cient, for each C as above and each assignment r : N R+ with R L! rn 0; that there exist dR-resolvable sets Ln Brn (cn) Brn (cn): Since L! 1 1 1 \ 1 Br (c) = x : dR(c ; x ) < r = x : dR(c ; y) < r and y = x = 1 f 1 g f g y : dR(c ; y) < r ; this is yet another condition relating inversion to f g R 1 1 the dR-topology, via the sets Br (c ) .

2 Analyticity: upper semicontinuity and the Baire property

This section enables some standard (separable) analyticity and category arguments to be lifted to the non-separable context. The three results we need are Hansell’s Characterization Theorem (Th. 2.1), yielding representation of analytic sets in the form j N H(j) with H upper semicontinuous and 2 compact-valued, Nikodym’sTheorem (Th. 2.2), implying their Baire prop- S erty, and the conclusion that analytic base--discrete maps (A--d maps, for short) are the only ones that matter (compare Prop. 2.9). We content ourselves mostly with a metric context, though a wider one is feasible (consult [Han-92]). Recall that a metric space S is said to be absolutely analytic, or just analytic, if it is Souslin- (S); i.e. is Souslin in its F (metric) completion S. A Hausdor¤ space S is -analytic if S = i I K(i) K 2 for some upper-semicontinuous map K from I to (S); the compact subsets K S of S: In a separable metric space, an absolutely analytic subset is -analytic K Analytic Baire Spaces 9

([Jay-Rog, Cor 2.4.3 plus Th. 2.5.3]). In a non-separable complete metric space X; it is not possible to represent a Souslin- (X) subset S of X as F a -analytic set relative to I = NN. Various generalizations of countability K now enter the picture, as we now recall, referring to two survey papers: [St2] and the more recent [Han-92]. Denoting by wt(X) the weight of the space X (i.e. the smallest cardi- nality of a base for the topology), and replacing I = NN by J = N for = wt(X); with basic open sets J(j n) := j0 J : j0 n = j n ; consider j f 2 j j g sets S represented by the following extended -Souslin operation (brie‡y: the extended Souslin operation)

S = H(j); where H(j) := H(j n); j J n N 2 2 j [ \ applied to a determining system H(j n) := H(j n): j N of sets from h j i h j 2 i a family subject to the requirement that: H (i) H(j n): j n n is -discrete for each n. f j j 2 g For = the corresponding extended Souslin- sets reduce to the H F F -Souslin sets of [Han-92]. (This slightly re…nes Hansell’sterminology, and abandons Stone’s term ‘-restricted Souslin’of [St2].) Say that the determining system is shrinking if n (ii) diamX H(j n) < 2 ; so that H(j) is empty or single-valued, and so j compact.

With X above complete (e.g. X = S) and for = (X); the mapping H : H F J (X) evidently yields a natural upper semi-continuous representation !K of S. We refer to it below, in relation to the Analytic Cantor Theorem, and also in Prop 2.9; there the fact that C := j : H(j) = is closed in f 6 ;g N yields a natural representation of S as the image of C under a map h de…ned by H(j) = h(j) : The map h is continuous and index--discrete f g with countable …bres (by (i) above), as noted in Remarks 1.5.1.

Theorem 2.1 (Characterization of Analytic sets, [Han-73a]). In a metric space X, the Souslin- (X) subsets of X are precisely the sets S F represented by a shrinking determining system of closed sets through the extended -Souslin representation above with = wt(X).

For other equivalent representations, including a weakening of -discreteness in X above to -d relative to its union, as well as to -d decompositions, see [Hans-73b] and [Han-73a]. Thus, working relative to J, the corresponding extended Souslin sets exhibit properties similar to the -analytic sets K relative to I: In particular of interest here is: 10 A. J. Ostaszewski

Theorem 2.2 (Nikodym Theorem for Analytic sets). In a metric space S, analytic sets have the Baire property.

Proof. Since S (S) = (S); the theorem follows immediately from \F F the de…nition of analytic sets as Souslin- (S) and from Nikodym’sclassical F theorem ([Jay-Rog, §2.9], or [Kech, Th. 29.14, cf. Th. 21.6]) asserting that the Baire property is preserved by the usual Souslin operation, with the consequence that Souslin- sets have the Baire property (since a closed set F di¤ers from its interior by a nowhere dense set).

Using Hansell’scharacterization theorem and again Nikodym’sclassical theorem, one also has the equally thematic result:

Theorem 2.3 (Nikodym Theorem for extended Souslin sets). In a metric space, sets with a shrinking extended Souslin- representation have F the Baire property.

Actually, this is a direct consequence of the following result, apparently unrecorded in the literature, so for completeness and in view of its brevity we include a proof (despite not needing it).

Theorem 2.4 (Nikodym Stability Theorem for the extended Souslin operation). In a topological space, the extended Souslin operation applied to a determining system of sets with the Baire property yields a set with the Baire property.

Proof. One need only check that the classical ‘separable’proof given for the usual Souslin operation in [Jay-Rog, Th. 2.9.2, pp. 43-4] continues to hold mutatis mutandis for the choice of the family of sets with the Baire M property and of the meagre subsets of the metric space. In particular, N we must interpret NN there as N throughout, with (N) denoting …nite sequence with terms in . With this our sole aim, assume the extended Souslin operation above is applied to a determining system of sets B( n) h j i in . So B( n): n n is a -d family for each n. By Banach’s M f j j 2 g localization principle, or Category Theorem ([Oxt] Ch. 16, [Kel] Th. 6.35, [Jay-Rog] p. 44-5, or [Kur-1] §10.III under the name Union Theorem), is N closed under -d unions, and hence so is (open sets being closed under M arbitrary union). As the system D( n) is a re…nement of the B( n) h j i h j i system, D( n): n n is also -d for each n, and so the union f j j 2 g D( n; t): t is in . (Note that the sets D( n) are de…ned as f j 2 g M j …nite intersections of sets in .) Each set N( n) := D( n) D( n; t): S M j j n f j S Analytic Baire Spaces 11 t is in , as is closed under subset formation, and again the 2 g N N family N( n): n n is -d for each n; as before by re…nement: f j j 2 g N( n) D( n): Hence j j L := N(): (N) = N( n): n n n N f 2 g 2 f j j 2 g is in , again[ by Banach’sCategory[ Theorem.[ The remainder of the proof N in [Jay-Rog, Th. 2.9.2, pp. 43-4] now applies verbatim.

A similar, but simpler, argument with the Radon measurable sets M shows these to be stable under the extended Souslin operation (using measure completeness and local determination, for which see [Fre4, 412J, cf. 431A], and measurable envelopes [Fre2, 213L]). Evidently, the standard separable category arguments may be also be applied to -d decompositions of a set, in view of Banach’sCategory Theorem, just cited. Finally, since H : J (X) above is upper-semicontinuous (for X com- !K plete and = ), the following theorem, used in the separable context of H F [Ost-AH, §2] and [Ost-LB3, Th. AC] continues to hold in the non-separable context (by the same proof), which permits us to quote freely some of its consequences in §4 in such a context.

Theorem 2.5 (Analytic Cantor Theorem). Let X be a Hausdor¤ space and A = K(J); with K : J (X) compact-valued and upper-semicontinuous. !K If Fn is a decreasing sequence of (non-empty) closed sets in X such that

Fn K(J(j1; :::; jn)) = ; for some j = (j1; :::) J and each n; then \ 6 ; 2 K(j) Fn = : \ n 6 ; T Here, beyond upper-semicontinuity, we do not need properties related to the notion of -d possessed by the mapping H (for which see [HJR]). We return to a discussion of analytic base--discrete maps, promised in §1. Recall their de…nition requires in addition to base--discreteness that, for any discrete family of analytic sets in X; the family f( ) has a -d E E base consisting of sets with the Baire property. The remaining results in this section are gleaned from a close reading of the main results in [Han-74] in respect of base--discrete maps, i.e. Hansell’s sequence of results 3.6- 3.10 and Th. 4.1, all of which derive the required base--discrete property by arguments that combine -d decompositions with discrete collections of singletons. We shall see below that all these results may be re…ned to the A--d context. 12 A. J. Ostaszewski

Theorem 2.6. An index--discrete function is A--d (Analytic base-- discrete).

Proof. Suppose that an indexed family := At : t T has a -d de- A f 2 g composition using / discrete families n := Atn : t T ; with At = Atn A f 2 g n for each t; and all the sets A have the Baire property (so in particular, t S if the sets are analytic –by Nikodym’s Theorem above (Th.2.2)). Putting

A~tn := At Atn and n := A~tn : t T ; which is discrete, we obtain a \ A f 2 g B--d decomposition for (or an A--d one in the case of an analytic -d A decomposition), and soe a fortiori a B--d base for (or an A--d one). A Thus, if is a discrete family of analytic sets, and f is index--discrete, E then := f( ), which comprises analytic sets (see Remark 1.5.2 above), A E has for its -d base the family n of analytic sets, so with the Baire n A property, where n are as just given above. A S e e Remark 2.7. The a--d maps are closed under composition (cf. [Han-74] Prop.3.4); also by the construction in Th. 2.6 above, the conclusions of [Han-74] Prop. 3.7, Cor. 3.8 and 3.9 yield A--d bases for f( ), when E E comprises analytic sets. In similar vein are the next two re…nements of results due to Hansell –together verifying the adequacy of A--d maps for analytic-sets theory.

Proposition 2.8 ([Han-74] Prop. 3.10). A closed surjective map onto a metrizable space is A--d.

Proof. Since singletons are analytic, a base that is a discrete family of singletons is an A--d base. This combined with the construction in Th. 2.6 above re…nes the argument for Prop. 3.10 proving that a closed surjective map onto a metrizable space is an A--d map.

Proposition 2.9 ([Han-74] Th. 4.1). Analytic metric spaces are the A--d continuous images of N.

Proof. By Remark 1.5.2 again, there is only one direction to consider. So let S be analytic; we re…ne Hansell’s argument. Observe …rst that the argument in for Prop. 3.5(ii) proves more: if Y is -discrete, then any map into Y is A--d, (as in Th. 2.6 above). Next, using the notation H for analytic sets established above (in a complete context, with = ); work H F in the closed subspace C N comprising those j with H(j) = ; and de…ne, 6 ; Analytic Baire Spaces 13 as above, the (continuous) map h on C via H(j) = h(j) : Observe that h f g takes, for each n; the discrete family of basic open sets J(j n); relativized to j C; to the -d family of analytic sets h(J(j n) C); and so h is an A--d map j \ (by [Han-74] Cor. 3.9, reported in Remark 1.5.4 above). Stone’s canonical retraction r of N onto any closed subspace as applied to the closed subspace C has -discrete range on N C and is the identity homeomorphism on C n –for details see [Eng-ret] (where r is also shown to be a closed map). So, in view of the preceding two observations, r is an A--d map. Hence h r is A--d, since composition of A--d maps is A--d and provides the required characterization.

3 Generalized Levi Theorem

The generalized Levi characterization of Baire spaces in Main Theorem 1.6 is a consequence of the following result, which we also apply in §6.

Lemma 3.1 (Generalized Levi Lemma). If f : X Y is surjective, con- ! tinuous and Baire base--discrete (in particular, index--discrete) from X metric and analytic to Y a paracompact space, then there is a dense metriz- 1 able -subspace Y 0 Y such that for X0 := f (Y 0) the restriction map G f X0 : X0 Y 0 is open. j !

Proof. Let = n = At : t T be an open base for X with n A n A f 2 g A = Atn : t Tn discrete. Then Et := f(At) is analytic (see Remarks 1.5.2), f 2 g S so has the Baire property (Th. 2.2, Nikodym’sTheorem for analytic sets).

Let n = f(A): A n and let n be a -d base for n comprising sets E f 2 A g B E witht the Baire property. Put n = nm with each nm discrete. Thus B m B B for each t T and Et n one has 2 2 E S

Et = B : B Et and B nm : m f 2 B g [ [ Put = nm. For each B pick an open set UB and meagre sets B nm B 2 B N and M such that B SB B = (UB NB) MB n [ with MB disjoint from MB and with NB UB: As B : B nm is f 2 B g discrete, the set

M := MB : B nm mn f 2 B g is meagre. By paracompactness[ of[Y (cf. [Eng, Th. 5.1.18]), since UB NB : f n B nm is discrete, for B nm we may select open sets WB with 2 B g 2 B 14 A. J. Ostaszewski

UB NB WB with WB : B nm discrete. Without loss of generality n f 2 B g WB UB (otherwise replace WB by WB UB): So UB NB WB UB and \ n hence UB NB = WB NB: Then n n

N := WB NB : B nm n;m f \ 2 B g [ [ is also meagre. Now put Y 0 := Y (M N) and W := WB : B n [ m;n f 2 nm ; which is open. Then, for B nm one has B g 2 B S S

B Y 0 = WB Y 0; \ \ so that B is open relative to Y 0 and also Y 0 WB : B nm is open and f \ 2 B g discrete in Y 0: Now for t T; since Et n for some n and n is a base for 2 2 E B n; E

Et Y 0 = B Y 0 : B Et & B nm \ n;m f \ 2 B g [ [ = WB Y 0 : B Et & B nm n;m f \ 2 B g [ [ is open in Y 0. For G X open, since is a (topological) base, we may write A G G G := n with n := A : A G & A n : n A A f 2 A g [ Then

G G f(G) := f( n ) with f( n ) := E : E = f(A) & A G & A n : n A A f 2 A g [ 1 So for X0 := f (Y 0)

G f(G X0) = Y 0 nm \ \ n;m W G with : = W[B : B f(A) & B nm & A G & A n ; Wnm f 2 B 2 A g which is open in Y 0:

Since f is continuous and nm := WB : B nm is discrete for each W f 2 B g m; n, this also shows that the family Y 0 WB : B nm is a - n;mf \ 2 B g d base for Y : Being paracompact, Y is regular ([Eng] Th. 5.1.5), so the 0 S subspace Y 0 is regular ([Eng] Th. 2.1.6), and so Y 0 is metrizable by Bing’s Characterization Theorem (see [Eng] Th. 4.4.8, cf. Remarks 1.5). Finally, by replacing the meagre sets M;N by larger sets that are unions of closed nowhere dense sets, we obtain in place of Y 0 a smaller, metrizable, dense

-subspace. G Analytic Baire Spaces 15

Remarks 3.2. 1. For Lemma 3.1 we replaced each system UB : B nm f 2 B g to obtain discrete systems WB : B nm to reduce the sets NB to the sets f 2 B g NB WB and only then did we take unions. This circumvents the suggested \ approach (in a parenthetical remark) to the proof of Th. 6.4(c) in [Han-92]

(by way of representing a Baire set as E = (GE PE) QE with PE E n [ (sic) for GE open and PE;QE meagre). 2. Given an arbitrary base one may replace each B with a Baire + B + + +2 B envelope B such that B B B: Then = n n is -d and Et + + B + B E := B : B Et and B n ; with E Et meagre. However, it t n f 2 B g t nS is not clear that the union of these meagre sets is meagre. S S

Proof of Main Theorem 1.6 (Non-separable Levi Open Mapping Theorem). Let X be analytic of weight : Then for some closed subset P N of there is a continuous index--discrete-map f : P X: Form P 0 ! and X0 analogously to X0 and Y 0 in the preceding lemma. As X is a Baire space and X0 is co-meagre, without loss of generality X0 is a dense and G N is metrizable. Also P 0 is a subspace of the complete space ; hence G is also topologically complete. So P 0 has the desired properties. As X0 is metrizable, the result now follows from Hausdor¤’sTheorem that the image under an open continuous mapping of the completely metrizable space P 0 onto a metrizable space X0 is also completely metrizable (for a proof see e.g. [Anc], or for a recent account e.g. [HP]).

For the converse, as X0 is metrizable, the result again follows from Haus- dor¤’sTheorem. Thus X0 is completely metrizable. But its complement in X is meagre. So X is a Baire space –in fact an almost complete space.

4 Normed group preliminaries

We recall the de…nition of a normed group from [BOst-N] and cite from from [Ost-LB3] four results that we need in the next two sections. The …rst (Th. 4.3) is quite general, but we need to observe here that in view of §2 the other three (Ths. 4.4, 4.6, 4.8) continue to hold in the new non-separable context here.

De…nition 4.1. For T an algebraic group (i.e. with no topology) with neutral element e, say that : T R+ is a group-norm ([BOst-N]) if jj jj ! the following properties hold: (i) Subadditivity (Triangle inequality): st s + t ; jj jj jj jj jj jj 16 A. J. Ostaszewski

(ii) Positivity: t > 0 for t = e and e = 0; jj jj 6 1 jj jj (iii) Inversion (Symmetry): t = t : jj jj jj jj Remarks 4.2. 1. The group-norm generates a right and a left norm topology (equivalently, right and left uniformity –cf. [Kel, Ch. 6 Pr. O]) via the T 1 T right-invariant and left-invariant metrics dR(s; t) := st and dL(s; t) := 1 T 1 1 jj jj s t = dR(s ; t ): We omit the superscript if context allows and identify jj jj 1 the two topologies by reference to the metric. Since dL(t; e) = dL(e; t ) = dR(e; t); convergence at e is identical under either topology. Evidently (T; dR) is homeomorphic with (T; dL) under inversion. As a result topological properties (e.g. density character) of the two norm topologies dR; dL are in fact norm properties, although properties of a subset of T (e.g. Baire property) may depend on the choice of norm topology. Preference of dR over dL is motivated by the right-invariance of the supremum metric on the space of bounded homeomorphisms –see [Ost-S]; in the absence of a quali…er, the ‘right’norm topology is to be understood. 2. Under either norm topology, there is continuity of operations at e: At further distances the topology may force the group operations to be in- creasingly ‘less’continuous.

3. Note that in the right norm-topology the right shift t(s) := st is a uniformly continuous homeomorphism, since dR(sy; ty) = dR(s; t); thus dR makes T a right-topological group.

4. Under the dR topology, Br(x) = t : dR(t; x) < r = Br(eT )x: T f g T 5. If d is a one-sidedly invariant metric on T , then t := d (t; eT ) is a jj jj norm.

Notation. We use the subscripts R; L; S as in xn R x etc. to indicate ! convergence in the corresponding metrics dR; dL; dS derived from the norm 1 (so that e.g. dR(x; y) := xy –see §1). Note that a metrizable topological jj jj group is a normed group, by the Birkho¤-Kakutani Theorem ([Bir], [Kak]; in fact this is a normability theorem for certain right-topological groups – see [Ost-LB3], or [Ost-S]).

Theorem 4.3 (Equivalence Theorem, [BOst-N, Th. 3.4]). A normed group X is a topological group under the right (resp. left) norm-topology i¤ each conjugacy 1 g(x) := gxg is right-to-right (resp. left-to-left) continuous at x = eX (and so every- Analytic Baire Spaces 17

where), i.e. for zn R eX and any g; ! 1 gzng R eX : ! Equivalently, it is a topological group i¤ left/right-shifts are continuous for the right/left norm topology, or i¤ the two norm topologies are themselves equivalent, i.e. the left and right uniformities generated by the norm coincide. The following results were proved in [Ost-LB3] for classically analytic spaces. Their proofs continue to hold for the more general non-separable de…nition of analytic space given and reviewed in §2, since those proofs in fact rely only on the Analytic Cantor Theorem as stated in Th. 2.5 above, and -d decomposition.

Theorem 4.4 ([Ost-LB3, Th. 1]). In a normed group X under dR; if X contains a non-meagre analytic set, then X is a Baire space. Remark 4.5. In the present normed-group context, in an almost-complete space (cf. §1), ‘Baire set’as ‘set with the Baire property’and ‘Baire space’ are almost-synonyms in the sense that: for B non-meagre, B has the Baire property i¤ B is a Baire space i¤ B is almost-complete (cf. [Ost-S, Th. 7.4]). Below ‘quasi all’means ‘all except for a meagre set of exceptions’. Theorem 4.6 (Analytic Shift Theorem, [Ost-LB3, Th. 3]). In a normed group under the topology dR, with zn eX and A analytic and non-meagre: ! for a non-meagre set of a A with co-meagre Baire envelope, there is an 2 in…nite set Ma and points an A converging to a such that 2 1 aam zmam : m Ma A: f 2 g In particular, if the normed group is topological, for quasi all a A, there 2 is an in…nite set Ma such that

azm : m Ma A: f 2 g

Remark 4.7. When zn eX one says that zn is a null sequence. Note 1 ! that aam zmam above also converges under dR to a as

1 1 1 1 1 dR(aa zmam; a) = aa zmama aa + zm + ama : m jj m jj jj m jj jj jj jj jj The theorem uses shifted-conjugacies to embed a subsequence of the null sequence into A; it is natural, borrowing from [Par], to term this a ‘shift- compactness’– see [Ost-LB3] for background and connections with allied notions of generic automorphisms. 18 A. J. Ostaszewski

Theorem 4.8 (Analytic Squared-Pettis-Theorem, [BOst-N, Th. 5.8]

). For X a normed group, if A is analytic and non-meagre under dR, then 1 2 eX is an interior point of (AA ) :

5 Non-separable automatic continuity of homomorphisms

In the proof of the Semi-Completeness Theorem (Main Theorem 1.9) we will need to know that the inverse of a continuous bijective homomorphism is continuous. In the separable case this follows by noting that the graph of the homomorphism is closed and, as a consequence of the Souslin-graph theorem, the inverse is a Baire homomorphism (meaning that preimages of open sets have the Baire property), and hence continuous. However, in the non-separable case the paradigm falls foul of the technical requirement for -discreteness. We will employ a modi…ed approach based on the following.

Theorem 5.1 (Open Homomorphism Theorem). For normed groups X;Y with X analytic and Y a Baire space, let f : X Y be a surjective, ! continuous homomorphism, which is base--discrete. Then f is open. So if 1 also X = Y and f is bijective, then f is continuous.

Proof. Suppose that B and A are arbitrary open balls around eX with B4 A: Let D be a dense set in X (e.g. X itself). Now = Bd : d D U f 2 g is an open cover of X: (Indeed, if x X and d D Bx; then x Bd; 2 2 \ 2 by symmetry.) Let be a -d open re…nement of , and say = n; V U V n V with each n discrete. Then, as X = ( V : V n ); we have Y = V n f 2 V g S ( f(V ): V n ): As f is base--discrete each n := f( n) = n f 2 V g S S W V f(V ): V n has a -d base n; write n := nm with each nm fS S 2 V g B B m B B discrete. So for each V n one has f(V ) := B nm : B f(V ) 2 V mfS 2 B g and so S Y = B nm : B f(V ) for some V n : nmf 2 B 2 V g [ As Y is non-meagre, there are n; m N;V n and B nm such that 2 2 V 2 B B f(V ) and B is non-meagre; for otherwise, since nm is discrete, by B Banach’s Category Theorem B nm : B f(V ) for some V n is f 2 B 2 V g meagre implying the contradiction that also Y is meagre. Pick such m; n and

B and V such that B f(V ) f(Bd) Now V Bd for some d D; as n 2 V re…nes , and so B f(V ) f(Bd) = f(B)f(d) is non-meagre. So f(B) is U non-meagre and analytic (as B is analytic). By the Squared Pettis Theorem Analytic Baire Spaces 19

1 2 1 1 1 1 (Th. 4.8), (f(B)(f(B)) ) = f(B)f(B) f(B)f(B) = f(BB BB ) is a neighbourhood of eY contained in f(A): The following corollary will be used together with the Lemma 1.10.

Theorem 5.2 (Continuous Inverse Theorem). If under dR the normed 1 group X is an analytic Baire space and the inversion map i : x x is ! -discrete preserving (takes discrete families to -discrete families), then the inverse of any continuous conjugacy ( ) is also continuous. x 1 Proof. If Vt : t T is -d, then for any x so is Vtx : t T ; as right f 2 g f 2 g shifts are homeomorphisms. Applying our assumption about the inversion 1 1 1 map, for any x the family xVt : t T is -d, hence xVt x : t T is f 1 2 g f 2 g -discrete, and so again xVtx : t T is -d. This means is continuous f 2 g x is index--discrete. By the preceding theorem (Th. 5.1), if x is continuous then its inverse is also continuous. With some minor amendments and from somewhat di¤erent hypothe- ses, the same proof as in the Open Homomorphism Theorem (Th. 5.1) demonstrates the following generalization of a separable result (given in [BOst-N] Th. 11.11), but unfortunately without any prospect for achieving the Baire property (see Remark 5.4 below). Here again the assumed discreteness preservation is ful…lled in the realm of separable spaces. We give the proof for the sake of comparison and because of its a¢ nity with a result due to Noll [N, Th. 1] concerning topological groups (not necessarily 1 metrizable), in which the map f has the property that f (U) is analytic for each open -set U. In our metric setting, when preimages under the F homomorphism f of open sets are analytic, f is Baire by Nikodym’s The- 1 orem and, since f ( ) is a disjoint and completely additive analytic for A discrete, the -d decomposability condition given below is satis…ed by A Hansell’sresult [Han-71] Th. 2 cited in Remarks 1.5.5. Noll shows the -d decomposability condition below is satis…ed when X is a topological group that is topologically complete (using the [Fh] generalization of Hansell’s result and of [KP] –cf. again Remarks 1.5.5). Theorem 5.3 (Baire Homomorphism Theorem). For normed groups X;Y with X analytic, a surjective Baire homomorphism f : X Y is 1 ! continuous provided f ( ) is -discretely decomposable for each -discrete A family in Y: A Proof. We proceed as above but now in Y . For " > 0; with B = B"=4(eY ) 1 open and D any dense set in Y choose ad with f(ad) = d: Put T := f (B); 20 A. J. Ostaszewski which has the Baire property (as f is Baire). As Y is metrizable, the open cover Bd : d D has a -d re…nement = n n with n := Atn : f 2 g A A A1 f t Tn discrete. For each n; by assumption, we may write f (Atn): t 2 g S f 2 Tn = Btnm : t Tn with Btnm : t Tn discrete in X for each m g nmf 2 g f 2 g and n: Now S 1 1 X = f (Y ) = f (Atn): t Tn = Btnm : t Tn : nf 2 g nmf 2 g [ [ As X is a Baire space, there are n; m such that Btnm : t Tn is f 2 g non-meagre. Again by Banach’s Category Theorem, and since B : t S tnm f 1 2 Tn is discrete, there is t with Btnm non-meagre. But Btnm f (Atn) g1 f (Bd) = T ad for some d D; as re…nes Bd : d D : Thus T ad and 2 A f 2 g so T is non-meagre, as the right shift ad is a homeomorphism. But T has the Baire property and X is analytic, so T contains a non-meagre analytic 1 2 subset. By the Squared Pettis Theorem (Th. 4.8) (TT ) contains a ball

B(eX ): Then 1 4 1 B(eX ) f [(B"=4) ] = f [B"(eY )]; proving continuity at eX :

Remark 5.4. In the separable case, by demanding that the graph of a homomorphism be Souslin- (X Y ) one achieves the Baire property of sets 1 F f (U); for U open in Y; by projection parallel to the Y -axis of (X U); \ provided that Y is a -analytic space. For Y absolutely analytic, one has K an extended Souslin representation, and hence a representation of Y as an upper-semicontinuous image of some product space N. But the proof of the projection theorem in [Jay-Rog, Ths.2.6.5 and 2.6.6] now yields that the projection of a Souslin- (X Y ) set has only a Souslin- (X) representation F F relative to N, without guaranteeing the -discreteness condition. In the non-separable context the Baire property can be generated by a projection theorem, provided one has both that the graph is absolutely analytic and that the relevant projection, namely (x; f(x)) x, is base--discrete (cf. ! [Han-92] Th. 4.6; [Han-74] §6, [Han-71] §3.5).

6 From normed to topological groups

In this section the generalized Levi result in Cor. 1.7 is the key ingredient; we will use it and results of earlier sections to prove the Semi-Completeness Theorem (Main Theorem 1.9). The proof layout (preparatory lemmas fol- lowed by proof) and strategy are the same as in [Ost-Joint], but, as some Analytic Baire Spaces 21 of the details di¤er, it is convenient to repeat the short common part (most of the proof of Lemma 6.2).

Lemma 6.1. For a normed group X; if (X; dS) is topologically complete and the continuous embedding map j :(X; dS) (X; dR) is Baire base-- ! discrete (e.g. A--d, in particular index--discrete), then there is a dense absolute- subset Y 0 in (X; dR) such that the restriction map j :(Y 0; dS) G ! (Y 0; dR) is open, and so all the three topological spaces (Y 0; dS) and (Y 0; dR) and (Y 0; dS) are homeomorphic (and topologically complete).

Proof. Take X = (X; dS) which is metric and analytic and Y = (X; dR) which is paracompact, then j : X Y is continuous, surjective and Baire ! base--discrete. By Lemma 3.1, there is a dense -subspace Y 0 of (X; dR) G 1 such that j Y 0 is an open mapping from (Y 0; dS) to (Y 0; dR): Now j (Y 0) is j a -subspace of (X; dS); so it follows that (Y 0; dS) is topologically complete, G so an absolute- . So j Y 0 embeds Y 0 as a subset of (X; dS) homeomorphi- G j cally into Y 0 as an absolute- subset of (X; dR): Since xn S x i¤ xn R x G ! ! and xn L x, all three topologies on Y 0 agree. !

Lemma 6.2. If in the setting of Lemma 6.1 the three topologies dR; dL; dS agree on a dense absolutely- set Y of (X; dR); then for any Y the 1 G 2 conjugacy (x) := x is continuous.

Proof. We work in (X; dR) which is thus analytic (as a base--discrete continuous image –Remarks 1.5.2). Let Y: We will …rst show that the 1 2 conjugacy x x is continuous in X at e; and then deduce that its ! 1 inverse x x is continuous. So let zn e be any null sequence in X. ! L ! Fix " > 0; then T := Y B" () is analytic and non-meagre, since X is a L\ Baire space (and Y B () is dR-open in Y with Y an absolute ). By \ " G the Analytic Shift Theorem (Th. 4.8), there are t T and tn in T with tn 2 1 converging to t (in dR) and an in…nite Mt such that ttm zmtm : m Mt f 21 g T: Since the three topologies agree on Y and the subsequence ttm zmtm of points of Y converges to t in Y under dR (see the Remark 4.7), the same 1 1 is true under dL: Using the identity dL(ttm zmtm; t) = dL(tm zmtm; e) = dL(zmtm; tm); we note that

1 t zmt = dL(t; zmt) dL(t; tm) + dL(tm; zmtm) + dL(zmtm; zmt) jj jj 1 dL(t; tm) + dL(tt zmtm; t) + dL(tm; t) 0; m ! 22 A. J. Ostaszewski

as m through Mt. So dL(t; zmt) < " for large enough m Mt. Then ! 1 2 for such m, as dL(; t) < ";

1 zm = dL(zm; ) dL(zm; zmt) + dL(zmt; t) + dL(t; ) jj jj dL(; t) + dL(t; zmt) + dL(t; ) 3": 1 Thus there are arbitrarily large m with zm 3": Inductively, taking jj jj 1 successively " = 1=n and k(n) > k(n 1) to be such that zk(n) 3=n; 1 jj jj one has zk(n) 0. By the weak continuity criterion (Lemma 3.5 of jj jj ! 1 [BOst-N], p. 37), (x) := x is continuous. Hence, by Lemma 1.10 and 1 Th. 5.2, (x) is also continuous.

Proof of Main Theorem 1.9 (Semi-Completeness Theorem). Un- der dR; the set Z := x : is continuous is a closed subsemigroup f x g of X ([BOst-N], Prop. 3.43, but using the Open Homomorphism Theo- rem, Th. 5.1 in place of the Souslin Graph Theorem). So as Y is dense,

X =clRY Z; i.e. x is continuous for all x; and so (X; dR) is a topolog- 1 1 ical group, by Th. 4.3. So xn R x i¤ x R x i¤ xn L x i¤ xn S x: ! n ! ! ! So (X; dR) is homeomorphic to (X; dS): Hence the topological group (X; dR) is topologically complete, being homeomorphic to (X; dS).

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