Proof of Product of Two Compact Spaces is Compact

By Compact Subset Criterion via Open Covers in the Ambient Space, it is enough to prove that every open cover of $X\times Y$ in $X\times Y$ has a finite subcover.

Let $I$ be a set, and let $(O_i)_{i\in I}$ be an open cover of $X\times Y$ in $X\times Y$, where $X\times Y$ is equipped with the product topology. Thus each $O_i$ is open in $X\times Y$ and

Fix a point $x\in X$. For each $y\in Y$, the point $(x,y)$ belongs to $X\times Y$, so there exists an index $i(x,y)\in I$ such that

Because $O_{i(x,y)}$ is open in the product topology, Product Topology gives open sets $U_{x,y}\subseteq X$ and $V_{x,y}\subseteq Y$ such that

As $y$ varies in $Y$, the family $(V_{x,y})_{y\in Y}$ is an open cover of $Y$ in $Y$. Since $Y$ is compact, Compact Subset Criterion via Open Covers in the Ambient Space applied to the subset $Y\subseteq Y$ yields a natural number $n_x\in\mathbb{N}$ and points $y_1,\dots,y_{n_x}\in Y$ such that

Set

Since each $U_{x,y_r}$ is open in $X$, the set $U_x$ is open in $X$ by Topological Space, and $x\in U_x$.

We claim that the finitely many sets

cover $U_x\times Y$. Let $(u,v)\in U_x\times Y$. Because the sets $V_{x,y_1},\dots,V_{x,y_{n_x}}$ cover $Y$, there exists some $r\in\{1,\dots,n_x\}$ such that $v\in V_{x,y_r}$. Since $u\in U_x\subseteq U_{x,y_r}$, we obtain

So the claim holds.

Now let $x$ vary in $X$. The family $(U_x)_{x\in X}$ is an open cover of $X$ in $X$. Since $X$ is compact, Compact Subset Criterion via Open Covers in the Ambient Space applied to the subset $X\subseteq X$ yields a natural number $m\in\mathbb{N}$ and points $x_1,\dots,x_m\in X$ such that

For each $s\in\{1,\dots,m\}$, the finitely many sets attached to $x_s$ cover $U_{x_s}\times Y$. Taking the union of all these finite families, we obtain finitely many members of the original family $(O_i)_{i\in I}$ that cover

Since

these finitely many members of $(O_i)_{i\in I}$ cover $X\times Y$. Thus the open cover $(O_i)_{i\in I}$ has a finite subcover.

Therefore every open cover of $X\times Y$ in $X\times Y$ has a finite subcover. By Compact Subset Criterion via Open Covers in the Ambient Space, the space $X\times Y$ is compact.