Katsuro Sakai

In this chapter, we introduce some terminology and notation. We also describe the standard Banach spaces contained in the product of real lines. Because of interest, some minor topics are also described even if those won’t be use hereafter.

This chapter provides advanced subjects concerning topological spaces for the reader who has finished the first course of General Topology. Those will be background knowledge to studying various branches of Topology. We derive some properties of the products of compact spaces and define Stone–Čech compactification. As generalization of localy compa...

In this chapter, several basic results on topological linear spaces and convex sets are presented. In order to provide background knowledge for Chapter 4, we will also detail matters that are omitted in an ordinary course on topological linear spaces. Here,we deal only with linear spaces over the real field R.We will characterize finite-dimensional...

In this chapter, we introduce and demonstrate the basic concepts and properties of simplicial complexes, where the local finiteness is not assumed. The importance and usefulness of simplicial complexes lay in the fact that they can be used to approximate and explore (topological) spaces. A polyhedron is the underlying space of a simplicial complex,...

For a collection A of subsets of a space 𝑋, the order of A is defined as follows: when 𝑛 = sup{cardA[𝑥] | 𝑥 ∈ 𝑋} is finite, we define ordA = 𝑛; otherwise, we write ordA = ∞. For an open cover U ∈ cov(𝑋), ordU = dim 𝑁(U) + 1, where 𝑁(U) is the nerve of U. The (covering) dimension of 𝑋, dim 𝑿, is defined as follows: dim 𝑋 ⩽ 𝑛 if each finite open cove...

This chapter is devoted to lectures on ANR Theory (Theory of Retracts). Basic properties and fundamental theorems are proved and various characterizations of ANRs and ANE(𝑛)s are givn. It is shown that the 𝑛-dimensional Menger compactum 𝜇 and Nöbeling space 𝜈 are shown to be AE(𝑛)s. It is proved that every 𝑛-dimensional metrizable space can be embe...

As mentioned in Preface, knowledge of homotopy groups is not required to read Sect. 4.20 but the second half of Chap. 7 requies a basic results for homotopy groups. We review here homotopy groups and characterize 𝑛-equivalences and weak homotopy equivalences in terms of homotopy groups. Moreover, we also prove some theorems which are required in th...

Preface and Contants of the Book: Geometric Aspect of General Topology

A compact set $A \not= \emptyset$ in $X$ is said to be {\bf cell-like} in $X$ if $A$ is contractible in every neighborhood of $A$ in $X$. A compactum $X$ is {\bf cell-like} if $X$ is cell-like in some {\it metrizable} space that contains $X$ as a subspace. It will be seen that $X$ is cell-like in every ANR that contains $X$ as a subspace (Theorem 7...

This chapter is written as preliminaries to study infinite-dimensional manifolds modeled on various spaces, e.g., Hilbert space, the Hilbert cube, incomplete pre-Hilbert spaces, the direct limit of Euclidean spaces, etc. The reader can recognize what kinds of knowledge are required. Almost all results contained in this chapter can be found in the a...

In this chapter, we prove the Torunczyk characterizations of Hilbert manifolds and Hilbert cube manifolds. By using the characterization of Hilbert space, we show that every Frechet space is homeomorphic to Hilbert space with the same weight. It is also proved that the space of maps from a non-discrete compactum to a complete metrizable separable A...

Preface and contents of the book "Topology of Infinite-Dimensional Manifolds"

This is Epilogue, References, and Index of the book "Topology of Infinite-Dimensional Manifolds."

In this chapter, we give fundamental results on manifolds modeled on Hilbert space (more generally an infinite-dimensional normed linear space) or the Hilbert cube.

In this chapter, we focus on proving the Triangulation Theorem for Hilbert cube manifolds. The compact case is bound up with the Borsuk conjecture asserting that every compact ANR has the homotopy type of a finite simplicial complex. The proof is based on Wall’s work on the homotopy type of a finite simplicial complex (Theorem 4.1.6) and a result o...

Let and ℝ∞Q∞ denote the direct limits of the following towers: ℝ⊂ℝ2⊂ℝ3⊂⋯;Q⊂Q2⊂Q3⊂⋯. As naturally expected, ℝ∞ is a locally convex topological linear space (Theorem 6.1.4). The space Q∞ is also homeomorphic to a locally convex topological linear space. In fact, it is known as the Heisey Theorem (Theorem 6.2.3) that Q∞ is homeomorphic to the dual spa...

In this chapter, we introduce (f.d.)cap sets in the separable Hilbert space (or the Hilbert cube) and characterize manifolds modeled on them. Then, we discuss their non-separable version called absorption bases, which are absolutely Fσ and homeomorphic one of the following spaces: ℓf2(Γ),ℓf2(Γ)×ℓQ2,ℓf2(Γ)×ℓ2,and;ℓ2(Γ)×ℓf2, where Γ is an infinite se...

In this Appendix, we give well-known characterizations of a combinatorial n-manifold. As one of them, we show that any triangulation of a PL n-manifold is a combinatrial n-manifold. It is also proved that a regular neighborhood of subpolyhedron of a PL n-manifold is also a PL n-manifold and if it does not meet boundary of the manifold, then the top...

An infinite-dimensional manifold is a topological manifold modeled on some infinite-dimensional homogeneous space called a model space. In this book, the following spaces are considered model spaces: Hilbert space (or non-separable Hilbert spaces), the Hilbert cube, dense subspaces of Hilbert spaces being universal spaces for absolute Borel spaces,...

The original version of the book has typos, incorrect symbols/characters, and small gaps in some proofs which have been fixed in the respective chapters of this book. Moreover, some results and remarks have also been added.

Let X be an infinite compact metrizable space having only a finite number of isolated points and Y be a non-degenerate dendrite with a distinguished end point v. For each continuous map ƒ : X → Y , we define the hypo-graph ↓vƒ = ∪ x∈X {x} × [v, ƒ (x)], where [v, ƒ (x)] is the unique arc from v to ƒ (x) in Y . Then we can regard ↓v C(X, Y ) = {↓vƒ |...

Let X be an infinite Peano space (i.e., locally compact, locally connected, separable metrizable space) and let Y be a 1-dimensional locally compact AR. The space of all continuous functions from X to Y with the compact-open topology is denoted by C(X,Y)C(X,Y). In this paper, we show that if X is non-discrete or Y is non-compact, then the function...

We prove that a topological group G is (locally) homeomorphic to an LF-space if G = boolean OR(n is an element of omega) G(n) for some increasing sequence of subgroups (G(n))(n is an element of omega) such that (1) for any neighborhoods U-n subset of G(n) , n is an element of omega, of the neutral element e is an element of G(n) subset of G, the se...

A subset A of a space X is called a retract of X if there is a map r : X→A such that r | A = id, which is called a retraction. As is easily observed, every retract of a space X is closed in X. A neighborhood retract of X is a closed set in X that is a retract of some neighborhood in X. A metrizable space X is called an absolute neighborhood retract...

In this paper, realizing an infinite simplicial complex K in the linear space RK(0) naturally, we investigate the box topology on |K| inherited from RK(0) that is finer than the metric topology and coarser than the Whitehead (weak) topology.

From the back cover of the book: “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 geom...

In this chapter, several basic results on topological linear spaces and convex sets are presented. We will characterize finite-dimensionality, metrizability, and normability of topological linear spaces. Among the important results are the Hahn–Banach Extension Theorem, the Separation Theorem, the Closed Graph Theorem, and the Open Mapping Theorem....

For an open cover \(\mathcal{U}\) of a space X, \(\mathrm{ord}\,\mathcal{U} =\sup \{ \mathrm{card}\,\mathcal{U}[x]\mid x \in X\}\) is called the order of \(\mathcal{U}\). Note that \(\mathrm{ord}\,\mathcal{U} =\dim N(\mathcal{U}) + 1\), where \(N(\mathcal{U})\) is the nerve of \(\mathcal{U}\). The (covering) dimension of X is defined as follows: di...

In this chapter, we are mainly concerned with metrization and paracompact spaces. We also derive some properties of the products of compact spaces and perfect maps. Several metrization theorems are proved, and we characterize completely metrizable spaces. We will study several different characteristics of paracompact spaces that indicate, in many s...

In this chapter, we introduce and demonstrate the basic concepts and properties of simplicial complexes. The importance and usefulness of simplicial complexes lies in the fact that they can be used to approximate and explore (topological) spaces. A polyhedron is the underlying space of a simplicial complex, which has two typical topologies, the so-...

The reader should have finished a first course in Set Theory and General Topology; basic knowledge of Linear Algebra is also a prerequisite. In this chapter, we introduce some terminology and notation. Additionally, we explain the concept of Banach spaces contained in the product of real lines.

A compact set A ≠ ∅ in X is said to be cell-like in X if A is contractible in every neighborhood of A in X. A compactum X is cell-like if X is cell-like in some metrizable space that contains X as a subspace. It will be seen that X is cell-like in every ANR that contains X as a subspace (Theorem 7.1.2). A cell-like (CE) map is a perfect (surjective...

The original version of the book has typos, incorrect symbols/characters, and small gaps in some proofs which have been fixed in the respective chapters of this book. Moreover, some results and remarks have also been added.

We prove that for any non-compact connected surface M the group Hc(M) of compactly suported homeomorphisms of M endowed with the Whitney topology is homeomorphic to R ∞ × l2 or Z × R ∞ × l2.

Let |K| be the metric polyhedron of a simplicial complex K. In this paper, we characterize a simplicial subdivision K ' of K preserving the metric topology for |K| as the one such that the set K '(0) of vertices of K ' is discrete in |K|. We also prove that two such subdivisions of K have such a common subdivision.

Let X be a Banach space and Conv(H) (X) be the space of non-empty closed convex subsets of X, endowed with the Hausdorff metric d(H). We prove that each connected component H of the space Conv(H) (X) is homeomorphic to one of the spaces: {0}, R, Rx (R) over bar (+), Q x (R) over bar (+), l(2), or the Hilbert space l(2)(k) of cardinality k >= c. Mor...

J. L. Taylor constructed a cell-like map of a compactum X onto the Hilbert cube I N such that X is not cell-like. In this note, we point out a defect in the construction and show how to fix it.

D. W. Henderson established the metric topology vertion of J. H. C. Whitehead's Theorem on small subdivisions of simplicial complexes. However, his proof is valid only for locally finite-dimensional simplicial complexes. In this note, we give a complete proof of Henderson's Theorem for arbitrary simplicial complexes.

Let F be a non-separable LF-space homeomorphic to the direct sum ΣnεN ℓ2(τn), where N 0 < τ1 < τ2 < .... It is proved that every open subset U of F is homeomorphic to the product ΙKΙ × F for some locally finite-dimensional simplicial complex K such that every vertex v ε K(0) has the star St(v, K) with card St(v, K)(0) < τ = sup τn (and card K(0) ≤...

In this paper, we study the group Hu(R) of uniform homeomorphisms having the uniform topology together with two subgroups H∞(R)={h∈Hu(R);lim|x|→ ∞ (h(x)-x)=0} and Hc(R), the group of compact support homeomorphisms. We show that the group Hu(R) is homeomorphic to ℓ∞×2א0 and that the triple (Hu(R)0,H∞(R),Hc(R)) is homeomorphic to (ℓ∞×ℓ2×ℓ2,{0}×ℓ2×ℓ2,...

For a non-compact n-manifold M let H(M) denote the group of homeomorphisms of M endowed with the Whitney topology and H_c(M) the subgroup of H(M) consisting of homeomorphisms with compact support. It is shown that the group H_c(M) is locally contractible and the identity component H_0(M) of H(M) is an open normal subgroup in H_c(M). This induces th...

We prove that for any non-compact connected surface $M$ the group $H_c(M)$ of compactly suported homeomorphisms of $M$ endowed with the Whitney topology is homeomorphic to $R^\infty\times l_2$ or $Z\times R^\infty\times l_2$.

Let X be a locally compact Polish space and G a non-discrete Polish ANR group. By C(X,G), we denote the topological group of all continuous maps f:X \to G endowed with the Whitney (graph) topology and by C_c(X,G) the subgroup consisting of all maps with compact support. It is known that if X is compact and non-discrete then the space C(X,G) is an l...

In this paper, we classify topologically the homeomorphism groups H(Γ) of infinite graphs Γ with respect to the compact-open and the Whitney topologies.

Let X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)ℕ of the Hilbert cube Q = [−1, 1]ℕ. In this paper, generalizing the Sakai-Uehara’s r...

Let X be a locally compact Polish space and G a non-discrete Polish ANR group. By C(X,G), we denote the topological group of all continuous maps f:X \to G endowed with the Whitney (graph) topology and by C_c(X,G) the subgroup consisting of all maps with compact support. It is known that if X is compact and non-discrete then the space C(X,G) is an l...

Let $CLB_H(X)$ denote the hyperspace of closed bounded subsets of a metric space $X$, endowed with the Hausdorff metric topology. We prove, among others, that natural dense subspaces of $CLB_H(R^m)$ of all nowhere dense closed sets, of all perfect sets, of all Cantor sets and of all Lebesgue measure zero sets are homeomorphic to the Hilbert space $...

Let X be an infinite-dimensional Banach space with density tau and let CCH (X) and Conv(H)(B) (X) be the spaces of all non-empty compact convex sets in X and of all non-empty bounded closed convex sets admitting the Hausdorff metric, respectively. In this note, it is proved that (i) CCH (X) is homeomorphic to l(2) (tau); (ii) if Conv(H)(B) (X) has...

Let ConvF(Rn) be the space of all non-empty closed convex sets in Euclidean space Rn endowed with the Fell topology. In this paper, we prove that ConvF(Rn) … Rn £ Q for every n > 1 whereas ConvF(R) …R £ I. Let Conv(X) be the set of all non-empty closed convex sets in a normed linear space X = (X;k¢k). We can consider various topologies on Conv(X)....

Let L(X) be the space of all lower semi-continuous extended real-valued func- tions on a Hausdorff space X, where, by identifying each f with the epi-graph epi(f), L(X) is regarded the subspace of the space CldF(X × R) of all closed sets in X × R with the Fell topology. Let LSC(X) = {f ∈ L(X) | f(X) ∩ R 6= ∅, f(X) ⊂ (−∞, ∞)} and LSCB(X) = {f ∈ L(X)...

For a metric space $X = (X,d)$ ,let $\mathrm{Cld}_H(X)$ be the space of all nonempty closed sets in $X$ with the topology induced by the Hausdorff extended metric: $$d_H(A,B) = \max\bigg\{\sup_{x\in B}d(x,A),\sup_{x\in A}d(x,B)\bigg\} \in [0,\infty].$$ On each component of $\mathrm{Cld}_H(X)$ , $d_H$ is a metric (i.e., $d_H(A,B) < \infty$ ). In thi...

Let X be a separable metric space. By CldW(X), we denote the hyperspace of non-empty closed subsets of X with the Wijsman topology. Let FinW(X) and BddW(X) be the subspaces of CldW(X) consisting of all non-empty finite sets and of all non-empty bounded closed sets, respectively. It is proved that if X is an infinite-dimensional separable Banach spa...

Let X be an infinite-dimensional Banach space with weight τ. By Cld AW (X), we denote the hyperspace of nonempty closed sets in X with the Attouch—Wets topology. By Fin AW (X), Comp AW (X) and Bdd AW (X), we denote the subspaces of Cld AW (X) consisting of finite sets, compact sets and bounded closed sets, respectively. In this paper, it is proved...

http://www.tulips.tsukuba.ac.jp/mylimedio/dl/page.do?issueid=652273&tocid=100082704&page=143-159

Let Cld AW (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 Cld AW (X) and its various subspaces are AR's. Moreover, if X is an infinite-dimensional Banach space with weight w(X) then Cld AW (X) is homeomorphic to a Hilbert space with weight 2 w(X).

By , we denote the space of all closed sets in a space X (including the empty set ∅) with the Fell topology. The subspaces of consisting of all compact sets and of all finite sets are denoted by and , respectively. Let Q=[−1,1]ω be the Hilbert cube, B(Q)=Q⧹(−1,1)ω (the pseudo-boundary of Q) and In this paper, we prove that is homeomorphic to (≈) Q...

By CldF(X), we denote the hyperspace of non-empty closed sets of a locally compact metrizable space X with the Fell topology. Let ContF(X) be its subspace consisting of all continua and ContF(X) the clo- sure of ContF(X) in CldF(X). It is proved that if X is connected, locally connected, non-compact and has no free arcs, then (ContF(X),ContF(X)) "...

In 1994, R. Cauty proved Geoghegan's conjecture on ANRs that a metrizable space X is an ANR if and only if every open set in X has the homotopy type of a CW-complex. In this note, by using the mapping cylinder technique, we provide an alternative short proof of this characterization of ANRs.

Let μn+1 be the (n+1)-dimensional universal Menger compactum. It is known that the n-shape category of Z-sets in μn+1 is isomorphic to the weak proper n-homotopy category of their complements. In this paper, we introduce the (n+1)-skeletal conic telescope to define the strong n-shape category of compacta, and show that the strong n-shape category o...

Let 1 ≤ p ≤ ∞. For each n-dimensional Banach space E = (E, ∥ · ∥), we define a norm ∥ · ∥p on E × R as follows: [formula] It is shown that the correspondence (E, ∥ · ∥) ↦ (E × R, ∥ · ∥p) defines a topological embedding of one Banach–Mazur compactum, BM(n), into another, BM(n 1), and hence we obtain a tower of Banach–Mazur compacta: BM(1) ⊂ BM(2)...

In this paper, considering the problem when the completion of a metric ANR $X$ is an ANR and $X$ is homotopy dense in the completion, we give some sufficient conditions. It is also shown that each uniform ANR is homotopy dense in any metric space containing $X$ isometrically as a dense subset, and that a metric space $X$ is a uniform ANR if and onl...

We identify Euclidean spaces Rn with the subspaces of the countable infinite product Rω . Then the set ⋃n∈NRn has two natural topologies, namely the weak topology (the direct limit) with respect to the tower R1⊂R2⊂R3⊂⋯ and the relative topology inherited from the product topology of Rω . We denote these spaces by R∞ and σ , respectively. Thus the b...

Let F∞(X) be the free topological semilattice over a kω -space X (i.e., the direct limit of a tower of compacta). It is proved that F∞(X) is homeomorphic to R∞=lim→Rn if and only if X is connected, X has no isolated points and every compactum in X is contained in a finite-dimensional locally connected compact metrizable subset of X . It is also sho...

http://www.tulips.tsukuba.ac.jp/mylimedio/dl/page.do?issueid=481178&tocid=100080738&page=69-80

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 of nonempty compact subsets of endowed with the Vietoris topology, where is the extended real li...

Let X = (X, d) be a metric space and let the product space X x R be endowed with the metric rho((x, t), (x',t')) = max{d(x,x'), \ t - t' \}. We denote by USCCB(X) the space of bounded upper semicontinuous multi-valued functions cp : X --> R such that each p(x) is a closed interval. We identify p epsilon USCCB(X) with its graph which is a closed sub...

In this paper, it is shown that the proper n-shape category of Ball-Sher type is isomorphic to a subcategory of the proper n-shape category defined by proper n-shaping. It is known that the latter is isomorphic to the shape category defined by the pair (Hpn, Hpn Pol), where Hpn is the category whose objects are locally compact separable metrizable...

In this paper, we describe the proper n-shape category by using non-continuous functions. Moreover, applying non-continuous homotopies, we show that the Cech expansion is a polyhedral expansion in the proper n-homotopy category. http://web.math.hr/glasnik/vol_33/no2_13.html

Let n be the n-dimensional universal Menger compactum, X a Z-set in n and G a metrizable zero-dimensional compact group with e the unit. It is proved that there exists a semi-free G-action on n such that X is the xed point set of every g 2 G r feg. As a corollary, it follows that each compactum with dim 6 n can be embedded in n as the xed point set...

Let P(X) be the space of probability measures on a space X and let P(X), P(X), P(X) and Pn(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, if and onl...

Generalizing the result of Chigogidze (1991), we prove that all autohomeomorphisms of connected Menger manifolds are stable (in the sense of Brown and Gluck).

It is proved that every separable zero-dimensional compact group acts freely on any Menger manifold M. In case M is compact, this result was proved by Dranishnikov. Here is provided an alternative short proof.

It is shown that any σ-compact metrizable space is an AR (ANR) if and only if it is (locally) equi-connected and has the compact (neighborhood) extension property.

It is shown that any sigma-compact metrizable space is an AR (ANR) if and only if it is (locally) equi-connected and has the compact (neighborhood) extension property.

Generalizing the result of Chigogidze (1991), we prove that all autohomeomorphisms of connected Menger manifolds are stable (in the sense of Brown and Gluck).

It is proved that every separable zero-dimensional compact group acts freely on any Menger manifold M. In case M is compact, this result was proved by A. N. Dranishnikov [Math. USSR, Izv. 32, No. 1, 217- 232 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 1, 212-228 (1988; Zbl 0648.57016)]. Here is provided an alternative short pro...

We prove a mapping theorem of Brown-Cassler type (Berlanga’s noncompact version) for manifolds modelled on the Hilbert cube Q and the (n+1)-dimensional universal Menger compactum μ n+1 .

Let X be a nondiscrete metric compactum and Y an Euclidean polyhedron without isolated points or a Lipschitz n-manifold (n > 0). Let LIP(X, Y)w be the space of Lipschitz maps from X to Y admitting the weak topology with respect to the tower 1-LIP(X, Y) ⊂ 2-LIP(X, Y) ⊂ 3-LIP(X, Y) ⊂ ⋯, where k-LIP(X, Y) denotes the space of Lipschitz maps with Lipsc...

It is shown that the hyperspace of a connected CW-complex is an absolute retract for stratifiable spaces, where the hyperspace is the space of non-empty compact (connected) sets with the Vietoris topology. 0. Introduction. The class S of stratifiable spaces (M3-spaces) con- tains both metrizable spaces and CW-complexes and has many desirable proper...

Singular homology is a beautiful theory, in which we can see a clear correspondence between Algebra and Topology. However, it behaves badly on topological spaces ... http://www.tulips.tsukuba.ac.jp/mylimedio/dl/page.do?issueid=185629&tocid=100000293&page=351-387

Let l2 denote a Hubert space, and let i2Q = {(xi) ∈ l2supi xi < ∞ and i2f = {xi,) ∈ l2xi = 0 except for finitely many i. We show that the triple (H(X), HLIP(X), HPL(X)) of spaces of homeomorphisms, of Lipschitz homeomorphisms, and of PL homeomorphisms of a finite graph X onto itself is an (l2, l2, i2f)-manifold triple, and that the triple (E(I, X),...

Let X be a compact PL manifold and Q denote the Hubert cube Iω. In this paper, we show that the following subgroups of the homeomorphism group H(X × Q) of X × Q are manifolds: In fact, let H*(X × Q) denote the subspace consisting of those homeomorphisms which are isotopic to a member of H*(X × Q), where * = fd, PL or LIP respectively. Then it is sh...

Let Q = [ -1, 1 ]? be the Hilbert cube and $Q_f = \{ (x_i) \in Q\mid x_i = 0\quad\text{except for finitely many}\quad i \}.$ For a compact connected polyhedron X with $\dim X > 0$, the hyperspaces of (nonempty) subcompacta, subcontinua, and compact subpolyhedra of X are denoted by 2X, C(X), and $\operatorname{Pol}(X)$, respectively. And let $C^{\op...