compactness is preserved under a continuous map
Theorem [1, 2] Suppose f:X→Y is a continuous map between topological spaces X and Y. If X is compact and f is surjective, then Y is compact.
The inclusion map [0,1]↪[0,2) shows that the requirement for f to be surjective cannot be omitted. If X is compact and f is continuous we can always conclude, however, that f(X) is compact, since f:X→f(X) is continuous (http://planetmath.org/IfFcolonXtoYIsContinuousThenFcolonXtoFXIsContinuous).
Proof of theorem. (Following [1].) Suppose {Vα∣α∈I} is an arbitrary open cover for f(X). Since f is continuous, it follows that
{f-1(Vα)∣α∈I} |
is a collection of open sets in X. Since A⊆f-1f(A) for any A⊆X, and since the inverse commutes with unions (see this page (http://planetmath.org/InverseImage)), we have
X | ⊆ | f-1f(X) | ||
= | f-1(⋃α∈I(Vα)) | |||
= | ⋃α∈If-1(Vα). |
Thus {f-1(Vα)∣α∈I} is an open cover for X. Since X is compact, there exists a finite subset J⊆I such that {f-1(Vα)∣α∈J} is a finite open cover for X. Since f is a surjection, we have ff-1(A)=A for any A⊆Y (see this page (http://planetmath.org/InverseImage)). Thus
f(X) | = | f(⋃i∈Jf-1(Vα)) | ||
= | ff-1⋃i∈Jf-1(Vα) | |||
= | ⋃i∈JVα. |
Thus {Vα∣α∈J} is an open cover for f(X), and f(X) 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 {Ai∣i∈I} is a collection of closed subsets of Y with the finite intersection property. Then {f-1(Ai)∣i∈I} is a collection of closed subsets of X with the finite intersection property, because if F⊆I is finite then
⋂i∈Ff-1(Ai)=f-1(⋂i∈FAi), |
which is nonempty as f is a surjection. As X is compact, we have
f-1(⋂i∈IAi)=⋂i∈If-1(Ai)≠∅ |
and so ⋂i∈IAi≠∅. Therefore Y 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 |