Theorem 2.2:

Suppose that $X$ is a compact topological space, $Y$ is an arbitrary Hausdorff space and $f : X \longrightarrow Y$ is a continuous surjection then

1. $Y$ is compact.
2. If $A$ is a closed subset of $X$ then $f(A)$ is closed.
3. It $f$ is bijective then $f$ is a homeomorphism.