Proof That the Continuous Image of a Compact Space is Compact

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Theorem

A continuous image of a compact set is compact.

Proof

Supposeiscontinuous andandaretopological spaces, andisa compact subset of

Letbean open cover ofsothat

Then

Sinceiscontinuous, the setsareopen andisan open cover of

iscompact, henceisreducible to a finite subcover, say

Then

Hence

Hence f(A) is compact.