Continuous Image of Compact Space is Compact

Theorem
Let $T_1$ and $T_2$ be topological spaces.

Let $f: T_1 \to T_2$ be a continuous mapping.

If $T_1$ is compact then so is its image $f \left[{T_1}\right]$ under $f$.

That is, compactness is a continuous invariant.

Proof
Suppose $\mathcal U$ is an open cover of $f \left[{T_1}\right]$ by sets open in $T_2$.

Because $f$ is continuous, it follows that $f^{-1} \left[{U}\right]$ is open in $T_1$ for all $U \in \mathcal U$.

The set $\left\{{f^{-1} \left[{U}\right]: U \in \mathcal U}\right\}$ is an open cover of $T_1$, because for any $x \in T_1$, it follows that $f \left({x}\right)$ must be in some $U \in \mathcal U$.

Because $T_1$ is compact, it has a finite subcover $\left\{{f^{-1} \left[{U_1}\right], f^{-1} \left[{U_2}\right], \ldots, f^{-1} \left[{U_r}\right]}\right\}$.

It follows that $\left\{{U_1, U_2, \ldots, U_r}\right\}$ is a finite subcover of $f \left[{T_1}\right]$.