We provide a machinery for transferring some properties of metrizable ANR-spaces to metrizable
LC
-spaces. As a result, we show that for completely metrizable spaces the properties ALC
, LC
and
WLC
...coincide to each other. We also provide the following spectral characterizations of ALC
and celllike
compacta: A compactum X is ALC
if and only if X is the limit space of a σ-complete inverse system S = {X
, p
, α < β < τ} consisting of compact metrizable LC
-spaces X
such that all bonding projections
p
, as a well all limit projections p
, are UV
-maps.
A compactum X is a cell-like (resp., UV
) space if and only if X is the limit space of a σ-complete inverse
system consisting of cell-like (resp., UV
) metrizable compacta.
Universal Cell-Like Maps Dydak, Jerzy; Mogilski, Jerzy
Proceedings of the American Mathematical Society,
1994, Volume:
122, Issue:
3
Journal Article
Peer reviewed
Open access
The main results of the paper are the following: Theorem. Suppose n ≤ ∞. 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' → Y1of (complete) separable metric spaces with dim X' ≤ n there exist (closed) embeddings$i: Y' \rightarrow Y$and$j: X' \rightarrow f^{-1}(i(Y'))$such that fj = if'. Corollary. Suppose$n < \infty$. There is a complete and separable metric space Y such that dimZY ≤ n, and any (complete) separable metric space Y' with dimZY' ≤ n embeds as a (closed) subset of Y.
Hereditarily aspherical compacta Dydak, Jerzy; Yokoi, Katsuya
Proceedings of the American Mathematical Society,
06/1996, Volume:
124, Issue:
6
Journal Article
Peer reviewed
Open access
{The notion of (strongly) hereditarily aspherical compacta introduced by Daverman (1991) is modified. The main results are: \begin{theorem1} If X\in LC^{1} is a hereditarily aspherical compactum, ...then X\in ANR. In particular, X is strongly hereditarily aspherical. \end{theorem1} \begin{theorem2} Suppose f:X\to Y is a cell-like map of compacta and f^{-1}(A) is shape aspherical for each closed subset A of Y. Then \begin{itemize} \item1. Y is hereditarily shape aspherical, \item2. f is a hereditary shape equivalence, \item3. \dim X\ge \dim Y. \end{itemize}\end{theorem2} \begin{theorem3} Suppose G is a group containing integers. Then the following conditions are equivalent: \begin{itemize} \item1. \dim X\le 2 and \dim _{G}X=1, \item2. \dim _{G*_{\Z }G}X=1. \end{itemize}\end{theorem3} \begin{theorem4} Suppose G is a group containing integers. If \dim X\le 2 and \dim _{G}X=1, then X is hereditarily shape aspherical. \end{theorem4} \begin{theorem5} Let X be a two-dimensional, locally connected and semilocally simply connected compactum. Then, for any compactum Y \begin{equation*}\dim (X \times Y) = \dim X + \dim Y.\end{equation*} \end{theorem5} }
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