The role of countable dimensionality in the theory of celllike relations
Author:
Fredric D. Ancel
Journal:
Trans. Amer. Math. Soc. 287 (1985), 140
MSC:
Primary 54C55; Secondary 54C56, 54C60, 54F45
MathSciNet review:
766204
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Abstract: Consider only metrizable spaces. The notion of a slicetrivial relation is introduced, and Theorem 3.2 is proved. This theorem sets forth sufficient conditions for a continuous relation with compact point images to be slicetrivial. 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 celllike map is a hereditary shape equivalence if there is a sequence of closed subsets of such that (1) is countable dimensional, and (2) is a hereditary shape equivalence for each . Theorem 5.9. If is a proper onto map whose point inverses are sets, then is an absolute neighborhood extensor for the class of countable dimensional spaces. Furthermore, if is countable dimensional, then is an absolute neighborhood retract. Theorem 5.9 is of particular interest when specialized to the identity map of a locally contractible space.
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DOI:
http://dx.doi.org/10.1090/S0002994719850766204X
PII:
S 00029947(1985)0766204X
Article copyright:
© Copyright 1985
American Mathematical Society
