Proof That the Continuous Image of a Compact Space is Compact
A continuous image of a compact set is compact.
Supposeiscontinuous andandaretopological spaces, andisa compact subset of
Letbean open cover ofsothat
Sinceiscontinuous, the setsareopen andisan open cover of
iscompact, henceisreducible to a finite subcover, say
Hence f(A) is compact.