Jerzy Mogilski

On topological classification of function spaces _{}() of low Borel complexity

We prove that if X is a countable nondiscrete completely regular space such that the function space Cp(X) is an absolute Fa-set, then Cp(X) is homeomorphic to v°°, where a = {(xi) E R°°:xi = O for all but finitely many i} . AS an application we answer in the negative some problems of A. V. Arhangelskil by giving examples of countable completely regular spaces X and Y such that X fails to be a bR-space and a k-space (and hence X is not a kc,,-space and not a sequential space) and Y fails to be an 80-space while the function spaces Cp(X) and Cp(Y) are homeomorphic to Cp(X) for the compact metric space 3S = {0} U {n1: n = 1, 2, . . . } .

Some applications of the topological characterizations of the sigma-compact spaces ²_{} and Σ

We use a technique involving skeletoids in cr-compact metric ARs to obtain some new examples of spaces homeomorphic to the cr-compact linear spaces I2, and E. For example, we show that (1) every No-dimensional metric linear space is homeomorphic to ¡?; (2) every cr-compact metric linear space which is an AR and which contains an infinite-dimensional compact convex subset is homeomorphic to E; and (3) every weak product of a sequence of cr-compact metric ARs which contain Hubert cubes is homeomorphic to E.

Universal cell-like maps

The main results of the paper are the following: Theorem. Suppose n < oo. There is a cell-like map f: X -Y of complete and separable metric spaces such that dim X < n, and for any cell-like map f : X -Y of (complete) separable metric spaces with dim X < n there exist (closed) embeddings i: Y -* Y and j: X -* f1(i(Y)) such that fi = if. Corollary. Suppose n < oo. There is a complete and separable metric space Y such that dimz Y < n, and any (complete) separable metric space Y with dimz Y < n embeds as a (closed) subset of Y.

Recent classification and characterization results in geometric topology

We announce a complete topological classification of the function spaces C (X) of Borel class not higher than 2, provided that I is a countable space. We also present a topological classification of the /c-dimensional universal pseudoboundaries and pseudointeriors in R , and we investigate under what conditions strong negligibility of crZ-sets characterizes Hilbert space manifolds.

Classification of finite-dimensional universal pseudo-boundaries and pseudo-interiors

Let n and k be fixed integers such that n ≥ 1 and 0 ≤ k ≤ n. Let B k n and s k n denote the k-dimensional universal pseudo-boundary and the k-dimensional universal pseudo-interior in R n , respectively. The aim of this paper is to prove that B k n is homeomorphic to B k m if and only if s k n is homeomorphic to s k m if and only if n=m or n, m ≥ 2k + 1

PROPERTY C AND FINE HOMOTOPY EQUIVALENCES

We show that within the class of metric a-compact spaces, proper fine homotopy equivalences preserve property C, which is a slight generalization of countable dimensionality. We also give an example of an open fine homotopy equivalence of a countable dimensional space onto a space containing the Hilbert cube.

Linear maps do not preserve countable dimensionality

Examples of linear maps between normed spaces are constructed, including a one-to-one map from a countable-dimensional linear subspace of 12 onto 12. We prove that the linear span of a countable-dimensional linearly independent subset of a normed linear space is, in many cases, countable dimensional.

Countable dimensionality and dimension raising cell-like maps

Abstract We show that countable dimensionality is not preserved under hereditary shape equivalences between complete spaces if such maps can make “arbitrarily large dimension jumps” inside the class of countably dimensional compacta.

An AR-map whose range is more infinite-dimensional than its domain

We construct an example of an AR-map f: X -Y, where X is a strongly countable dimensional compact AR and Y is a countable dimensional AR which is not strongly countable dimensional. Using this map we find a shrinkable decomposition of the pre-Hilbert space 12 whose quotient map does not stabilize to a near homeomorphism. We also present a partial result concerning the question whether cell-like maps preserve countable dimensionality.

Universal finite-to-one map and universal countable dimensional spaces

Abstract There is a closed finite-to-one map tf of a zero-dimensional, separable, metric absolute Gδσ-set X onto a space Y such that for any closed, finite-to-one map f′:X′ → Y′ of separable, metric spaces, with dim X′ ⩽ 0, there exist embeddings i : X′ → X and j : Y′ → Y such that fi = jf. In particular, the space Y is universal for all separable metric spaces which are countable dimensional. We also show that finite-to-one maps produce naturally cell-like maps. Finally, using the method of absorbers we prove a topological characterization of the space σ × N>, where σ is Smirnovs universal strongly countable dimensional space and N is Nagatas universal countable dimensional space.