compactness is preserved under a continuous map

Theorem [1, 2] Suppose f:XY is a continuous mapMathworldPlanetmath between topological spacesMathworldPlanetmath X and Y. If X is compactPlanetmathPlanetmath and f is surjectivePlanetmathPlanetmath, then Y is compact.

The inclusion mapMathworldPlanetmath [0,1][0,2) shows that the requirement for f to be surjective cannot be omitted. If X is compact and f is continuousMathworldPlanetmath we can always conclude, however, that f(X) is compact, since f:Xf(X) is continuous (

Proof of theorem. (Following [1].) Suppose {VααI} is an arbitrary open cover for f(X). Since f is continuous, it follows that


is a collectionMathworldPlanetmath of open sets in X. Since Af-1f(A) for any AX, and since the inversePlanetmathPlanetmathPlanetmath commutes with unions (see this page (, we have

X f-1f(X)
= f-1(αI(Vα))
= αIf-1(Vα).

Thus {f-1(Vα)αI} is an open cover for X. Since X is compact, there exists a finite subset JI such that {f-1(Vα)αJ} is a finite open cover for X. Since f is a surjection, we have ff-1(A)=A for any AY (see this page ( Thus

f(X) = f(iJf-1(Vα))
= ff-1iJf-1(Vα)
= iJVα.

Thus {VααJ} 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 (

Shorter proof. Suppose {AiiI} is a collection of closed subsets of Y with the finite intersection property. Then {f-1(Ai)iI} is a collection of closed subsets of X with the finite intersection property, because if FI is finite then


which is nonempty as f is a surjection. As X is compact, we have


and so iIAi. Therefore Y is compact.


  • 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