Some other convex-valued selection theorems 147 (2) Every lower semicontinuous mapping F : X! IR such that for every x 2 X, F (x) is either convex and

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1 PART B: RESULTS x1. CHARACTERIZATION OF NORMALITY- TYPE PROPERTIES 1. Some other convex-valued selection theorems In this section all multivalued mappings are assumed to have convex values in some Banach space. We begin with the union of Theorems (1.1) and (1.5) [258, Theorem (3.2)"]. Theorem (1.1). Let X be a T 1 -space. Then the following assertions are equivalent: (1) X is paracompact and (2) For every Banach space B, every lower semicontinuous map f : X! B with closed convex values admits a singlevalued continuous selection. It follows from the proof that the following is equivalent to properties (1) and (2) of Theorem (1.1): (3) For every set A, every lower semicontinuous map f : X! `1(A) with closed convex values admits a singlevalued continuous selection. Suppose that we replace in (2) the class of all Banach spaces by one of its subclasses, e.g. the class of all separable Banach spaces, or the class of all reexive (Hilbert, quasireexive, etc.) Banach spaces. Moreover, we can substitute in (2) the family of all closed convex sets by some of its subclasses, e.g. the class of all compacta, the class of all closed bounded sets, etc. The problem is to nd a suitable replacement for (1) such that the equivalence between (1) and (2) will be preserved. Some such questions have complete answers, others have partial or no answer. Each armative answer gives a selection characterization of some class of topological spaces. The classical Urysohn extension theorem asserts that the normality of a T 1 -space X is equivalent to the statement that the real line IR isanextension space for X, i.e. that for every closed subspace A X, every continuous map f : A! IR can be extended to a continuous map ^f : X! IR. But an extension problem is nothing but a special case of some selection problem. Hence, the class of all normal spaces is a natural candidate for some suitable substitution in (1). Theorem (1.2). Let X be a T 1 -space. Then the following assertions are equivalent: (1) X is normal 146

2 Some other convex-valued selection theorems 147 (2) Every lower semicontinuous mapping F : X! IR such that for every x 2 X, F (x) is either convex and compact or F (x) = IR, admits a singlevalued continuous selection and (3) For every separable Banach space B, every lower semicontinuous mapping F : X! B such that for every x 2 X, F (x) is either convex and compact or F (x) =B, admits a singlevalued continuous selection. The following theorem is an analogue of Theorem (1.2), without the separability condition. Theorem (1.3). Let X be a T 1 -space. Then the following assertions are equivalent: (1) X is collectionwise normal and (2) For every Banach space B, every lower semicontinuous mapping F : X! B such that for every x 2 X, F (x) is either convex and compact or F (x) =B, admits a singlevalued continuous selection. In the following theorem the separability condition has been added to the formulation of Theorem (1.1). Theorem (1.4). Let X be a T 1 -space. Then the following assertions are equivalent: (1) X is normal and countably paracompact (2) Each lower semicontinuous mapping F : X! IR with closed convex values admits a singlevalued continuous selection and (3) For every separable Banach space B, every lower semicontinuous mapping F : X! B with closed convex values admits a singlevalued continuous selection. Of course, it suces to consider only `1 instead of an arbitrary separable Banach space in Theorems (1.2) and (1.4) in the assertion (3). Theorems (1.2), (1.3), and (1.4) correspond to Theorems (3.1)', (3.2)', and (3.1)" of [258]. Also, it should be noted that with the substitution of extension properties instead of selection properties in Theorems (1.1) and (1.2) we obtain the classical characterization of normality [107,167], and respectively of collectionwise normality [101]. Finally, we recall that if each countable covering of a space X has a locally nite renement, then X is called a countably paracompact space and if for each disjoint, locally nite family ff g of closed subsets of X, there exists a disjoint family fg g of open subsets of X such that F G, for all, then X is said to be collectionwise normal. If such property holds for families ff g of 0, then X is said to be -collectionwise normal. Theorem (1.5) [85]. Let X be a T 1 -space and let be any cardinal number. Then the following assertions are equivalent: (1) X is -collectionwise normal and 147

3 148 Characterization of normality-type properties (2) For every Banach space B of weight, each lower semicontinuous mapping F : X! B such that for every x 2 X, F (x) is either convex and compact or F (x) =B, admits a singlevalued continuous selection. This theorem is interesting not only due to its relation to Theorem (1.3). Nedev observed that the original proof of the implication (1))(2) in Theorem (1.3) is valid only for compact-valued mappings F. Coban and Valov [85] gave the rst complete proof of Theorem (1.3), based on the method of coverings. More precisely, they found a compact-valued lower semicontinuous selection G for a mapping F from Theorem (1.3)(2). Consequently, the original Michael's proof works for conv G F. Hence, we can conclude that the method of covering sometimes looks more universal than the method of outside approximations for the case of convex-valued mappings. The method of coverings also plays a crucial role in selection theorems which \unify" Theorems (1.2){(1.4) with Zero-dimensional selection theorem (see x2, below). We shall conclude this section by a selection theorem for non- -paracompact domains, due to Nedev [303]: Theorem (1.6). Let be the space of all countable ordinals, endowed with the order topology, and B a reexive Banach space. Then every lower semicontinuous mapping F :! B with closed convex values, admits a singlevalued continuous selection. It is still an open problem whether this theorem characterizes countable paracompactness and collectionwise normality, in the spirit of Theorems (1.1){(1.4). 2. Characterizations via compact-valued selection theorems We list some of Coban's results [75,76]. If in the denition of paracompactness (See Theory, x1.1) one replaces the local niteness by the pointwise niteness of coverings then one obtains the denition of weak paracompactness. Theorem (1.7). For every regular space X, the following assertions are equivalent: (1) X is weakly paracompact and (2) For every completely metrizable space Y, every closed-valued lower semicontinuous mapping F : X! Y admits a compact-valued lower semicontinuous selection G : X! Y, i.e. G(x) F (x), for every x 2 X. Theorem (1.8). For every T 1 -space X, the following assertions are equivalent: (1) X is normal and (2) For every separable metrizable space Y, every compact-valued lower semicontinuous mapping F : X! Y admits a compact-valued upper semicontinuous selection. 148

4 Characterizations via compact-valued selection theorems 149 A space Y is said to be -paracompact (for an innite cardinal ) ifevery open covering of Y of cardinality has a locally nite open renement. Theorem (1.9). For every T 1 -space the following assertions are equivalent: (1) X is normal and -paracompact and (2) For every completely metrizable space Y of weight, every closed- -valued lower semicontinuous mapping F : X! Y admits a compact- -valued upper semicontinuous selection. Under the dimensional restriction on the domain X there exists a version of the last two theorems in which the compactness of values of the selection is replaced by a suitable niteness condition. Theorem (1.10). For every T 1 -space X, the following assertions are equivalent: (1) X is normal and dimx n and (2) For every separable metrizable space Y, every compact-valued lower semicontinuous mapping F : X! Y admits an upper semicontinuous selection G : X! Y, with values G(x) of cardinality at most n +1. Theorem (1.11). For every T 1 -space X, the following assertions are equivalent: (1) X is normal and -paracompact with dim X n and (2) For every completely metrizable space Y of weight, every closed- -valued lower semicontinuous mapping F : X! Y admits an upper semicontinuous selection G : X! Y with values G(x) with cardinality at most n +1. Nedev [305] noticed that the property of a lower semicontinuous closedand convex-valued mapping F : X! Y to have an upper semicontinuous closed-valued selection also yields a characterization of paracompactness of X. Coban and Nedev [84] have obtained characterizations of -collectionwise normality which generalize Theorem (1.8) and reads just like Theorem (1.9) with \normal and -paracompact" replaced by \-collectionwise normal" and \closed-valued F " replaced by \F (x) is compact or F (x) =Y ". Finally, let us mention that Compact-valued selection theorem also holds for a normal (not necessary paracompact) domain X and for a continuous closed-valued mapping into completely metrizable spaces [75]. In other words, one of the assumptions of Compact-valued selection theorem admits a weakening with a simultaneous strengthening of the other assumption. 149

5 150 Characterization of normality-type properties 3. Dense families of selections. Characterization of perfect normality Theorem (1.1) states that under certain conditions, a multivalued mapping F has at least one singlevalued continuous selection. Consider now the following construction. Choose a nite subset K of the domain X, choose arbitrary points y = y(x) 2 F (x), x 2 K, dene a lower semicontinuous mapping F K by F K (x) = fy(x)g F (x) x 2 K x 2 XnK and then, by means of Theorem (1.1), nd a singlevalued selection f K of F K. Such a map f K will be a selection of the multivalued mapping F having prescribed values of the xed nite subset of the domain. And if we change K over the family of all nite subsets of the domain X, we obtain a suciently large family of selections of a given lower semicontinuous mapping F. A more careful technique which generalizes the idea above to countable subsets and separable ranges, yields the following theorem: Theorem (1.12) [258]. If X is perfectly normal, B is a separable Banach space and F : X! B is a lower semicontinuous mapping with closed convex values, then there exists a countable family S of selections of F such that ff(x) j f 2 Sg is a dense subset of the value F (x), for every x 2 X. Every metric space is perfectly normal, whereas the converse is false. The following strengthening of Theorem (1.12) was proved in [264] for metric domains: Theorem (1.13). Let X be ametric space, B abanachspace, and F : X! B a lower semicontinuous mapping with closed convex values. Then for each innite cardinal, there exists a family S of selections of F with card(s) such that the set ff(x) j f 2 Sg is dense in F (x), whenever x 2 X, and F (x) has a dense subset of cardinality. Let us return to Theorem (1.12) and let F : X! B be a convex-valued (in general, nonclosed-valued) lower semicontinuous mapping. Applying Theorem (1.12) to the mapping Cl(F ) : X! B, we nd some countable family S of its selections. Fix an enumeration of the family S, say S = = fs 1 s 2...g, and consider the following selection s of the mapping Cl(F ): s(x) = 1X i=1 s i (x)=2 i x 2 X: Clearly, s is continuous and s(x) 2 Cl(F (x)), due to the convexity of F (x). However, sometimes s(x) 2 F (x). In this way it is possible to construct a selection for some mappings with nonclosed convex values. More precisely, if C is a closed, convex subset of a Banach space then a face of C is a closed convex subset D C such that each segment in C, which has an interior 150

6 Dense families of selections. Characterization of perfect normality 151 point ind, must lie entirely in D theinside of C is the set of all points in C which do not lie in any faceofc. Denition (1.14) [258]. A convex subset C of a Banach space is said to be of convex D-type if it contains all interior points of its closure. Examples of convex D-type sets are: (a) closed convex sets (b) convex subsets which contain at least one interior point (in the usual metric sense) (c) nite dimensional convex sets (d) the subset of all strongly increasing functions on the interval [0 1] in the Banach space of all continuous functions on [0 1]. Before stating the next theorem, we note that if each closed subset of a space X is a G -subset, then X is called perfectly normal. Theorem (1.15) [258]. Let X be a T 1 -space. Then the following assertions are equivalent: (a) X is perfectly normal (b) Each convex-valued lower semicontinuous mapping F : X! IR n admits a singlevalued continuous selection and (c) For every separable Banach space B, each lower semicontinuous mapping F : X! B such that F (x) is a convex D-type subset of B, for all x 2 X, admits a singlevalued continuous selection. For an application of Theorem (1.15) in the theory of locally trivial brations, see Applications, x2.2. Let us recall (see Theory, x6) another selection theorem for nonclosed-valued multivalued mappings. Theorem (1.16). Let G be an open subset of a Banach space. Then each convex-valued lower semicontinuous mapping F from a paracompact space X into G with closed (in G) values F (x), x 2 X, admits a singlevalued continuous selection. Proof. First, we nd a compact-valued lower semicontinuous selection of F, say. Next, we consider the mapping = conv(). It easy to see, that : X! B is compact-valued (since the closed convex hull conv K in a Banach space is compact whenever K is compact), convex-valued and lower semicontinuous selection of the map F, and (x) F (x), for all x 2 X. Therefore, Convex-valued selection theorem can be applied to the map. Problem (1.17). Is it possible to substitute the set G in Theorem (1.16) by any G -subset of a Banach space? This is an interesting open problem in the theory of selections (see [275]). Here the main technical problem is the following: If B is a Banach space, G oneofitsg -subsets, F aconvex, closed (in G) subset of G and K acompact subset of F, then, in general, conv(k) is not necessarily a subset of F. The inclusion conv(k) F holds if, for example, G is the intersection of some countable family of open convex subsets of B e.g. Theorem (1.16) holds for the pseudo-interior of the Hilbert cube. 151

7 152 Characterization of normality-type properties A partial armative answer to the problem above was given by Gutev [161]. Theorem (1.18). Let X be a countably-dimensional metric space or a strongly countably-dimensional paracompact space. Then each convex-valued lower semicontinuous mapping F from X into a G -subset G of a Banach space B with closed (in G) values admits a singlevalued continuous selection. Note that in the theory of measurable multivalued mappings a countable dense (in the spirit of Theorem (1.12)) family of measurable selections is often called the Castaing representation of a given multivalued mapping (see x6, below). A special case was proved in [89] for a mapping F from a separable metric space X with a nite regular Borel measure into the unit ball D of the reexive Banach space L p (), for some 1 < p < 1 and some measure. Consider D endowed with the weak topology w. Then every Borel singlevalued mapping h : X! R D has the integral X hd, i.e. the unique point y 2 L p with Ay = Z X (A h)d for every A 2 L p. Let R be the mapping which associates to every Borel mapping h : X! D its integral R X hd. Theorem (1.19). Let F : X! D be a convex-valued lower semicontinuous mapping with F (x) being closed subsets of (D w) and let B F (resp. C F )be the family of all Borel (resp. continuous) singlevalued selections of F. Then R (CF ) is a dense subset of R (B F ), with respect to the norm topology. 4. Selections of nonclosed-valued equi-lc n mappings In this section we consider nonclosed-valued mappings. We include below results related to the weakening of the closedness of values F (x) in comparison with Theorems (1.15), (1.16) and (1.18) of Section 3. First, we note that the analogue of Problem (1.17) has an obvious armative solution with the substitution of Zero-dimensional selection theorem instead of Convex-valued ones. In fact, every G -subset G of a completely metrizable space Y is also completely metrizable. So, this selection theorem is directly applicable to the mapping F : X! G, G Y. In [274] it was shown that such a replacementis possible in the case of nite-dimensional selection theorem with simultaneous weakening of the condition that ff (x)g x2x is an equi-lc n family. The main point here is the following \factorization" idea. Let : X! Z be a mapping which satises the hypotheses of some selection theorem and hence has a selection ' : X! Z. Let h : Z! Y be a continuous mapping and F = h :X! Y. Then for the mapping F one has the obvious selection h' : 152

8 Selections of nonclosed-valued equi-lc n mappings 153 X! Y. But, on the other hand, the mapping F has in general no standard \selection" properties: closedness of F (x), n-connectedness of F (x), ELC n property for ff (x)g x2x, etc. If one can nd for a given F : X! Y such a representation F = h, then a selection theorem with weaker assumptions will be automatically valid. Moreover, it suces to have only F h, i.e. that h is a selection of F. It seems that rst such observation is due to Eilenberg [259, Footnote 10]. Denition (1.20). A mapping F : X! Y is said to be equi-lc n if the family ffxgf (x)g x2x is an equi-lc n family of subsets of the Cartesian product X Y. Every mapping F : X! Y with equi-lc n family ff (x)g x2x of values is equi-lc n, but the converse is false. For example, let X =IN, Y =IR, and F (m) =f0 1 m gir. Then F is ELCn -mapping, for every n 2 IN, but the family ff (m)g m2in is not an ELC 0 -family. Theorem (1.21) [259]. Let X be an (n +1)-dimensional metric space, Y a completely metrizable space and F : X! Y an ELC n lower semicontinuous mapping with closed values. Then F admits a continuous singlevalued selection. Proof. Finite-dimensional selection theorem can be applied to the mapping F from X into the (metric) completion of X Y,where F (x) =fxg F (x). It then suces to observe that F = p Y F. The key point here is that the product of two metrizable spaces is again metrizable. Note, that the product of a paracompact space and a metrizable space need not be paracompact. Theorem (1.22). Under the hypotheses of Theorem (1.21) let G be a G -subset of X Y, and replace the condition that F (x) are closed in Y by the condition that fxgf (x) are closed subsets of G. Then F admits a continuous singlevalued selection. Proof. One can consider F : X! G as a mapping into a completely metrizable space G,where G is a G -subset of the completion of X Y such that G \ (X Y )=G. A natural question arises whether Theorem (1.21) is true for arbitrary paracompact (nonmetrizable) domains? This problem was solved in [274]. Theorem (1.23). Finite-dimensional selection theorem can be strengthened simultaneously in two directions: (a) The assumption that ff (x)g x2x is ELC n family can be weakened to the assumption that F is ELC n mapping and (b) The assumption that F (x) are closed in Y, for every x 2 X, can be weakened to the assumption that there exists a G -subset G of X Y such that fxgf (x) are closed in G, for every x 2 X. 153

9 154 Characterization of normality-type properties Proof. We describe only how the mapping F can be factorized through the completely metrizable space Y (0 1] IN. Fix a representation G = = T 1 n=1 G n with G n an open subset of X Y and x a representation G n = = S fu n V n j 2 A n g as a union of \rectangular" sets, where A n is an index set. For every x 2 X and y 2 F (x), let ' n (x y) = supft >0 jfxgd(y t) U n V n for some 2 A ng : Finally, forevery x 2 X, let (x) =n fygf(0 'n (x y)]g 1 n=1 j y 2 F (x)o Y (0 1] IN : Clearly, F = p Y, where p Y : Y (0 1] IN! Y is the projection onto the rst factor. The rest of the proof is concerned with the verication that : X! Y (0 1] IN satises all the hypotheses of the standard Finite- -dimensional selection theorem. We complete this section by a remark that universality of Zero-dimensional selection theorem together with Theorem (1.23) gives the following \weak" Compact-valued selection theorem (see Theory, x4). Theorem (1.24). Compact-valued selection theorem remains valid if the assumption that F (x) are closed in Y, for every x 2 X, is weakened to the assumption that there exists a G -subset G of X Y such that fxgf (x) are closed in G, for every x 2 X. 154