Infinite-dimensional manifolds related to C-spaces
by
Oryslava Shabat
Lviv Academy of Printing

A topological space X has property C (briefly is a C-space) if for each sequence {a_n | n ∈ N} of open coverings of X there exists a sequence {b_n, n ∈ N} of open disjoint families such that each family b_n refines a_n and the family ∪_i=1^∞ b_n is a cover of X (see [1]). The transfinite dimension dim_C is introduced by Borst [2]. It is proved in [2] that a compact metrizable space X is a C-space if and only if dim_C(X) < w₁.

T. Radul [3] proved that for every countable ordinal a there exists a compact metrizable C-space X which contains a topological copy of every compact metrizable space Y with dim_C(Y) ≤ a.

This result allows us to construct, for a cofinal subset A of w₁, and every a ∈ A a k_w-space O_a which can be considered as a model space of the theory of infinite-dimensional manifolds. In the class of compact metrizable spaces X with dim_C(Y) < a, the O_a-manifolds play the role corresponding to that of R^∞-manifolds in the class of finite-dimensional compact metrizable spaces.

[1] D.F. Addis and J.H. Gresham, A class of infinite dimensional spaces. Part 1: Dimension theory and Alexandroff’s problem, Fund. Math. 101 (1978), 195-205.

[2] P. Borst, Some remarks concerning C-spaces, Preprint.

[3] T. Radul, Absorbing spaces for C-compacta, Topology and its Applications 83 (1998), 127-133.

Date received: May 8, 2008