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

Author:
Fredric D. Ancel

Journal:
Trans. Amer. Math. Soc. **287** (1985), 1-40

MSC:
Primary 54C55; Secondary 54C56, 54C60, 54F45

DOI:
https://doi.org/10.1090/S0002-9947-1985-0766204-X

MathSciNet review:
766204

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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 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* *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:
https://doi.org/10.1090/S0002-9947-1985-0766204-X

Article copyright:
© Copyright 1985
American Mathematical Society