Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 766204
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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\vert{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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54C55, 54C56, 54C60, 54F45

Retrieve articles in all journals with MSC: 54C55, 54C56, 54C60, 54F45


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0766204-X
PII: S 0002-9947(1985)0766204-X
Article copyright: © Copyright 1985 American Mathematical Society