continuous image of a compact set is compact

Theorem 1.

The continuousPlanetmathPlanetmath image of a compact set is also compact.


Let X and Y be topological spacesMathworldPlanetmath, f:XY be continuous, A be a compact subset of X, I be an indexing set, and {Vα}αI be an open cover of f(A). Thus, f(A)αIVα. Therefore, Af-1(f(A))f-1(αIVα)=αIf-1(Vα).

Since f is continuous, each f-1(Vα) is an open subset of X. Since AαIf-1(Vα) and A is compact, there exists n with Aj=1nf-1(Vαj) for some α1,,αnI. Hence, f(A)f(j=1nf-1(Vαj))=f(f-1(j=1nVαj))j=1nVαj. It follows that f(A) is compact. ∎

Title continuous image of a compact set is compact
Canonical name ContinuousImageOfACompactSetIsCompact
Date of creation 2013-03-22 15:53:14
Last modified on 2013-03-22 15:53:14
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 16
Author Wkbj79 (1863)
Entry type Theorem
Classification msc 54D30
Related topic CompactnessIsPreservedUnderAContinuousMap