# Proof regarding compact $T_2$-spaces and closed continuity.

Question: Prove that any continuous function from a compact $T_2$-space onto a $T_2$-space is closed, that is, $f(F)$ is closed if $F$ is closed.

Is my general reasoning correct?

Any compact subset of a $T_2$ space is closed. Also, A subset of a compact $T_2$-space is compact iff it is closed.

Further, compactness is preserved by continuous functions. Since compactness is preserved by continuous functions and the codomain is a $T_2$-space, then the image is also closed. Thus, any continuous function from a compact $T_2$-space onto a $T_2$-space is closed.