Theorem
A continuous image of a compact set is compact.
Proof
Supposeiscontinuous and
and
aretopological spaces, and
isa compact subset of
Letbean open cover of
sothat
Then
Sinceiscontinuous, the sets
areopen and
isan open cover of
iscompact, hence
isreducible to a finite subcover, say
Then
Hence
Hence f(A) is compact.