The role of countable dimensionality in the theory of cell-like relations

Abstract:

Consider only metrizable spaces. The notion of a slice-trivial relation is introduced, and Theorem 3.2 is proved. This theorem sets forth sufficient conditions for a continuous relation with compact $U{V^\infty }$ point images to be slice-trivial. Theorem 4.5 posits a number of necessary and sufficient conditions for a map to be a hereditary shape equivalence. Several applications of these two theorems are made, including the following. Theorem 5.1. A cell-like map $f:X \to Y$ is a hereditary shape equivalence if there is a sequence $\{ {K_n}\}$ of closed subsets of $Y$ such that (1) $Y - \bigcup \nolimits _{n = 1}^\infty {{K_n}}$ is countable dimensional, and (2) $f|{f^{ - 1}}({K_n}):{f^{ - 1}}({K_n}) \to {K_n}$ is a hereditary shape equivalence for each $n \geq 1$. Theorem 5.9. If $f:X \to Y$ is a proper onto map whose point inverses are $U{V^\infty }$ sets, then $Y$ is an absolute neighborhood extensor for the class of countable dimensional spaces. Furthermore, if $Y$ is countable dimensional, then $Y$ is an absolute neighborhood retract. Theorem 5.9 is of particular interest when specialized to the identity map of a locally contractible space.