In mathematics, in the field of topology, a topological space is said to be a shrinking space if every open cover admits a shrinking. A shrinking of an open cover is another open cover indexed by the same indexing set, with the property that the closure of each open set in the shrinking lies inside the corresponding original open set.[1]
YouTube Encyclopedic
-
1/1Views:4 703
-
From Delhi — Pervez Hoodbhoy in conversation with Siddharth Varadarajan on Pak-India politics
Transcription
Properties
The following facts are known about shrinking spaces:
- Every shrinking space is normal.[1]
- Every shrinking space is countably paracompact.[1]
- In a normal space, every locally finite, and in fact, every point-finite open cover admits a shrinking.[1]
- Thus, every normal metacompact space is a shrinking space. In particular, every paracompact space is a shrinking space.[1]
These facts are particularly important because shrinking of open covers is a common technique in the theory of differential manifolds and while constructing functions using a partition of unity.
See also
- Topological property – Mathematical property of a space
References
- ^ a b c d e Hart, K. P.; Nagata, Jun-iti; Vaughan, J. E. (2003), Encyclopedia of General Topology, Elsevier, p. 199, ISBN 9780080530864.
- General topology, Stephen Willard, definition 15.9 p. 104
 | This topology-related article is a stub. You can help Wikipedia by expanding it. |
This page was last edited on 27 November 2022, at 08:03