compactness is preserved under a continuous map
Theorem [1, 2] Suppose is a continuous map between topological spaces and . If is compact and is surjective, then is compact.
The inclusion map shows that the requirement for to be surjective cannot be omitted. If is compact and is continuous we can always conclude, however, that is compact, since is continuous (http://planetmath.org/IfFcolonXtoYIsContinuousThenFcolonXtoFXIsContinuous).
Proof of theorem. (Following [1].) Suppose is an arbitrary open cover for . Since is continuous, it follows that
is a collection of open sets in . Since for any , and since the inverse commutes with unions (see this page (http://planetmath.org/InverseImage)), we have
Thus is an open cover for . Since is compact, there exists a finite subset such that is a finite open cover for . Since is a surjection, we have for any (see this page (http://planetmath.org/InverseImage)). Thus
Thus is an open cover for , and is compact.
A shorter proof can be given using the characterization of compactness by the finite intersection property (http://planetmath.org/ASpaceIsCompactIfAndOnlyIfTheSpaceHasTheFiniteIntersectionProperty):
Shorter proof. Suppose is a collection of closed subsets of with the finite intersection property. Then is a collection of closed subsets of with the finite intersection property, because if is finite then
which is nonempty as is a surjection. As is compact, we have
and so . Therefore is compact.
References
- 1 I.M. Singer, J.A.Thorpe, Lecture Notes on Elementary Topology and Geometry, Springer-Verlag, 1967.
- 2 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
- 3 G.J. Jameson, Topology and Normed Spaces, Chapman and Hall, 1974.
Title | compactness is preserved under a continuous map |
---|---|
Canonical name | CompactnessIsPreservedUnderAContinuousMap |
Date of creation | 2013-03-22 13:55:50 |
Last modified on | 2013-03-22 13:55:50 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 13 |
Author | yark (2760) |
Entry type | Theorem |
Classification | msc 54D30 |
Related topic | ContinuousImageOfACompactSpaceIsCompact |
Related topic | ContinuousImageOfACompactSetIsCompact |
Related topic | ConnectednessIsPreservedUnderAContinuousMap |