Katsuro Sakai

Geometric aspects of general topology

This book is designed for graduate students to acquire knowledge of dimension theory, ANR theory (theory of retracts), and related topics. These two theories are connected with various fields in geometric topology and in general topology as well. Hence, for students who wish to research subjects in general and geometric topology, understanding these theories will be valuable. Many proofs are illustrated by figures or diagrams, making it easier to understand the ideas of those proofs. Although exercises as such are not included, some results are given with only a sketch of their proofs. Completing the proofs in detail provides good exercise and training for graduate students and will be useful in graduate classes or seminars. Researchers should also find this book very helpful, because it contains many subjects that are not presented in usual textbooks, e.g., dim X x I = dim X + 1 for a metrizable space X; the difference between the small and large inductive dimensions; a hereditarily infinite-dimensional space; the ANR-ness of locally contractible countable-dimensional metrizable spaces; an infinite-dimensional space with finite cohomological dimension; a dimension raising cell-like map; and a non-AR metric linear space. The final chapter enables students to understand how deeply related the two theories are. Simplicial complexes are very useful in topology and are indispensable for studying the theories of both dimension and ANRs. There are many textbooks from which some knowledge of these subjects can be obtained, but no textbook discusses non-locally finite simplicial complexes in detail. So, when we encounter them, we have to refer to the original papers. For instance, J.H.C. Whiteheads theorem on small subdivisions is very important, but its proof cannot be found in any textbook. The homotopy type of simplicial complexes is discussed in textbooks on algebraic topology using CW complexes, but geometrical arguments using simplicial complexes are rather easy.

Hyperspaces of non-compact metrizable spaces which are homeomorphic to the Hilbert cube☆

Abstract By Cld ∗ F (X) , we denote the space of all closed sets in a space X (including the empty set ∅) with the Fell topology. The subspaces of Cld ∗ F (X) consisting of all compact sets and of all finite sets are denoted by Comp ∗ F (X) and Fin ∗ F (X) , respectively. Let Q=[−1,1]ω be the Hilbert cube, B(Q)=Q⧹(−1,1)ω (the pseudo-boundary of Q) and Q f ={(x i ) i∈ N ∈Q∣x i =0 except for finitely many i∈ N }. In this paper, we prove that Cld ∗ F (X) is homeomorphic to (≈) Q if and only if X is a locally compact, locally connected separable metrizable space with no compact components. Moreover, this is equivalent to Comp ∗ F (X)≈B(Q) . In case X is strongly countable-dimensional, this is also equivalent to Fin ∗ F (X)≈Q f .

On R∞ and Q∞-manifolds

Abstract We give a characterization of manifolds modeled on R ∞ = dir lim or R n Q ∞ =dir lim Q n , where Q is the Hilbert cube, and elementary short proofs of the Open Embedding Theorem for these manifolds and the following theorem generalizing the Stability Theorem: Each fine homotopy equivalence between these manifolds is a near homeomorphism . Moreover we establish the Open Embedding Approximation Theorem.

Hyperspaces of Normed Linear Spaces with the Attouch–Wets Topology

Let CldAW(X) be the hyperspace of nonempty closed subsets of a normed linear space X with the Attouch–Wets topology. It is shown that the space CldAW(X) and its various subspaces are ARs. Moreover, if X is an infinite-dimensional Banach space with weight w(X) then CldAW(X) is homeomorphic to a Hilbert space with weight 2w(X).

A Hilbert cube compactification of the Banach space of continuous functions

Abstract Let C(X) be the Banach space of continuous real-valued functions of an infinite compactum X with the sup-norm, which is homeomorphic to the pseudo-interior s = (−1, 1)ω of the Hilbert cube Q = [−1, 1]ω. We can regard C(X) as a subspace of the hyperspace exp(X × R ) of nonempty compact subsets of X × R endowed with the Vietoris topology, where R = [−∞, ∞] is the extended real line (cf. (Fedorchuk, 1991)). Then the closure R (X) of C(X) in exp(X × R ) is a compactification of C(X). We show that the pair ( C (X), C(X)) is homeomorphic to (Q, s) if X is locally connected. As a corollary, we give the affirmative answer to a question of Fedorchuk (Fedorchuk, 1996, Question 2.6).

Spaces of measures on metrizable spaces

Abstract Let P ( X ) be the space of probability measures on a space X and let P S ( X ), P R ( X ), P Z ( X ) and P n ( X ) be subspaces of P ( X ) consisting of measures with separable supports, compact supports, finite supports and with supports consisting of at most n points, respectively. It is shown that, for a quadruple ( T , X , Y , Z ) of separable metrizable spaces, (P(T), P(X), P R (Y), P Z (Z)) ≈ (Q, s, Σ, σ) if and only if T is compact, X ⊂ ≠ T is G δ in T , Y is open in T , Y ∼ N and Z is σ-fd-compact and dense in T , where ≈ means “is homeomorphic to”, Q = [−1, 1] ω is the Hilbert cube, s = (−1, 1) ω the pseudo-interior of Q , Σ = {(x i ) ϵ Q ¦ sup ¦x i ¦ the radial-interior of Q and σ = {(x i ) ϵ s ¦ x i = 0 except for finitely many i} . In case X is nonseparable, we prove that P S (X) is homeomorphic to a Hilbert space l 2 ( A ) if and only if X is completely metrizable and dens X = card A (⪢ ℵ 0 ) . Moreover it is proved that ( P S (l 2 ( A )), P Z (l 2 ( A ))) ≈ (l 2 ( A ) ω , l 2 ( A ) ω f ) and P n (l 2 ( A )) ≈ l 2 ( A ) for an arbitrary infinite set A and that P Z ( E ) ≈ P n ( E ) ≈ E for any pre-Hilbert space E which is homeomorphic to E ω f = {(x i ) ϵ E ω ¦ x i = 0 except for finitely many i ϵ N } ⊂ E ω . We have similar results for the space M + ( X ) of nonnegative measures of X .

On infinite-dimensional manifold triples

Let Q denote the Hilbert cube [-1, 1]w, s = (-1, 1)w the pseudo-interior of Q, = {(xi) e sl sup xi < I} and = {(xi) e sIxi = 0 except for finitely many i} . A triple (X, M, N) of separable metrizable spaces is called a (Q, X, a)(or (s, X, c)-)manifold triple if it is locally homeomorphic to (Q, X, a) (or (s, X, a)). In this paper, we study such manifold triples and give some characterizations.

The space of Lipschitz maps from a compactum to a locally convex set

Abstract Let X be a non-discrete metric compactum and Y a separable locally compact, locally convex set without isolated points in a normed linear space. The spaces of continuous maps and Lipschitz maps from X to Y are denoted by C ( X , Y ) and LIP( X , Y ), respectively. For each k >0, k -LIP( X , Y ) and LIP k ( X , Y ) denote the subspaces of all f ; ϵ LIP( X , Y ) with the Lipschitz constant lip f ; ⩽ k and lip f k , respectively. Here is proved that ( C ( X , Y ), LIP( X , Y )) is an ( s , Σ )-manifold pair and each LIP k ( X , Y ) is a Σ -manifold, where s = (−1, 1) ω is the pseudo-interior of the Hilbert cube Q = [−1, 1] ω and Σ = {( x i ) ϵ Q | sup | x i | k -LIP( X , Y ) need not be a Q -manifold. However, in case Y is open in its convex hull, it is proved that ( k -LIP( X , Y ), LIP k ( X , Y )) is a ( Q , Σ )-manifold pair for each k >0.

Combinatorial infinite-dimensional manifolds and R∞-manifolds

Abstract Generalizing combinatorial manifolds to the infinite-dimensional case, we can define combinatorial ∞-manifolds. And then we can see that each R ∞-manifold is triangulated by a combinatorial ∞-manifold. The main purpose of this paper is to prove the Hauptvermutung for combinatorial ∞-manifolds: Any two homeomorphic combinatorial ∞-manifolds are combinatorially equivalent, that is, they have simplicially isomorphic subdivisions. As an application, we have the stable Hauptvermutung for simplicial complexes: For any homeomorphic countable simplicial complexes K and L, the product complexes K × Δ∞ and L × Δ∞ are combinatorially equivalent, where Δ∞ is the countably infinite full complex.

Semi-free actions of zero-dimensional compact groups on Menger compacta

Let μn be the n-dimensional universal Menger compactum, X a Z-set in μn and G a metrizable zero-dimensional compact group with e the unit. It is proved that there exists a semi-free G-action on μn such that X is the fixed point set of every g ∈ G r {e}. As a corollary, it follows that each compactum with dim 6 n can be embedded in μn as the fixed point set of some semi-free G-action on μn. In [Dr], Dranishnikov showed that every metrizable zero-dimensional compact group G acts freely on the n-dimensional universal Menger compactum μ (cf. [Sa]). Here we consider the fixed point sets of semi-free actions of G on μ. A closed set X in μ is called a Z-set if there are maps f : μ → μ r X arbitrarily close to id. The following is our result: Theorem. Let G be a metrizable zero-dimensional compact group with e the unit and X a Z-set in μ. Then there exists a semi-free G-action on μ such that X is the fixed point set of every g ∈ G r {e}. By [Be, 2.3.8], each compactum X with dimX 6 n can be embedded in μ as a Z-set. Then we have the following: Corollary. Let G be a metrizable zero-dimensional compact group. Each compactum X with dimX 6 n can be embedded in μ as the fixed point set of some semi-free G-action on μ. In the proof below, for two simplicial complexes K and L, K × L denotes the simplicial complex defined as the barycentric subdivision of the cell complex { σ×τ ∣∣ σ ∈ K, τ ∈ L}. For any simplicial map f : K → L, the simplicial mapping cylinder of f is denoted by M(f) (cf. [Wh, §6]). Notice that K and L are subcomplexes of M(f). By K, we denote the set of vertices (0-skeleton) of K. Proof of Theorem. We may only consider the case that G is non-trivial, i.e., G 6= {e}. By a well-known theorem of Pontryagin [Po, §46, C], G is the inverse limit of Received by the editors April 16, 1994 and, in revised form, April 28, 1996. 1991 Mathematics Subject Classification. Primary 54F15, 54H25, 54H15; Secondary 57S10, 22C05.