Proof of Product of Two Compact Spaces is Compact

By Compact Topological Space and Compact Subset §compact we must show that for every subset $\mathcal{O}$ of $\mathcal{T}$ with $X\times Y\subseteq\bigcup\mathcal{O}$ there is a finite subset of $\mathcal{O}$ whose union contains $X\times Y$. Fix such a set $\mathcal{O}$. The sets $\mathcal{T}_{X}$ and $\mathcal{T}_{Y}$ are topologies on $X$ and on $Y$, and $(X,\mathcal{T}_{X})$ and $(Y,\mathcal{T}_{Y})$ are compact by hypothesis.

Step 1 (a tube around one point of $X$). Fix $x\in X$, and let $\mathcal{W}_{x}$ be the set of all $V\in\mathcal{P}(Y)$ for which there are $U\in\mathcal{T}_{X}$ with $x\in U$ and a finite subset $\mathcal{O}'$ of $\mathcal{O}$ with $U\times V\subseteq\bigcup\mathcal{O}'$. If $V\in\mathcal{W}_{x}$ and $V'\subseteq V$, then $V'\in\mathcal{W}_{x}$, with the same $U$ and $\mathcal{O}'$, because $U\times V'\subseteq U\times V$.

(a) The open members of $\mathcal{W}_{x}$ cover $Y$. Let $y\in Y$. Then $(x,y)\in X\times Y\subseteq\bigcup\mathcal{O}$, so $(x,y)\in O$ for some $O\in\mathcal{O}$. As $O\in\mathcal{T}$, Product Topology §product-topology gives $U\in\mathcal{T}_{X}$ and $V\in\mathcal{T}_{Y}$ with $x\in U$, $y\in V$ and $U\times V\subseteq O$. The set $\{O\}$ is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice §small, it is a subset of $\mathcal{O}$, and $U\times V\subseteq O=\bigcup\{O\}$; hence $V\in\mathcal{W}_{x}\cap\mathcal{T}_{Y}$ and $y\in V$. Thus $Y\subseteq\bigcup(\mathcal{W}_{x}\cap\mathcal{T}_{Y})$.

(b) Finite unions. For $n\in\mathbb{N}_{0}$, with $[n]$ as in Intervals of Natural Numbers §segment, let $P(n)$ be the assertion that $\bigcup\mathcal{W}\in\mathcal{W}_{x}$ for every subset $\mathcal{W}$ of $\mathcal{W}_{x}$ onto which there is a bijection from $[n]$. We prove $P(n)$ for every $n\in\mathbb{N}_{0}$ by induction from $0$, The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §induction. For $n=0$ we have $[0]=\emptyset$ by Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting §segment, so a set $\mathcal{W}$ onto which there is a bijection from $[0]$ is the image of $\emptyset$, that is $\mathcal{W}=\emptyset$ and $\bigcup\mathcal{W}=\emptyset$; the set $X$ lies in $\mathcal{T}_{X}$ by Topological Space §empty-whole and contains $x$, the set $\emptyset$ is a finite subset of $\mathcal{O}$ by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice §small, and $X\times\emptyset=\emptyset\subseteq\bigcup\emptyset$, so $\emptyset\in\mathcal{W}_{x}$. Now assume $P(n)$, and let $\beta:[n+1]\to\mathcal{W}$ be a bijection onto a subset $\mathcal{W}$ of $\mathcal{W}_{x}$. By Intervals of Natural Numbers: Initial Segments, Adding One Element, Splitting and Shifting §successor, $[n+1]=[n]\cup\{n+1\}$ and $n+1\notin[n]$. Put $\mathcal{W}'=\beta[[n]]$ and $V=\beta(n+1)$. Then $\mathcal{W}=\mathcal{W}'\cup\{V\}$ by Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §union, and the restriction $\beta|_{[n]}$ is a bijection from $[n]$ onto $\mathcal{W}'$, being injective as $\beta$ is and mapping onto its image. By $P(n)$ there are $U'\in\mathcal{T}_{X}$ with $x\in U'$ and a finite subset $\mathcal{O}'$ of $\mathcal{O}$ with $U'\times\bigcup\mathcal{W}'\subseteq\bigcup\mathcal{O}'$; since $V\in\mathcal{W}_{x}$, there are $U''\in\mathcal{T}_{X}$ with $x\in U''$ and a finite subset $\mathcal{O}''$ of $\mathcal{O}$ with $U''\times V\subseteq\bigcup\mathcal{O}''$. Then $U'\cap U''\in\mathcal{T}_{X}$ by Topological Space §intersections, $x\in U'\cap U''$, the set $\mathcal{O}'\cup\mathcal{O}''$ is a finite subset of $\mathcal{O}$ by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice §union, and $\bigcup\mathcal{W}\in\mathcal{P}(Y)$ because $\mathcal{W}\subseteq\mathcal{W}_{x}\subseteq\mathcal{P}(Y)$. As $U'\cap U''\subseteq U'$, we have $(U'\cap U'')\times\bigcup\mathcal{W}'\subseteq U'\times\bigcup\mathcal{W}'\subseteq\bigcup\mathcal{O}'$, and likewise $(U'\cap U'')\times V\subseteq U''\times V\subseteq\bigcup\mathcal{O}''$; hence

$$(U'\cap U'')\times\bigcup\mathcal{W}=\Bigl((U'\cap U'')\times\bigcup\mathcal{W}'\Bigr)\cup\Bigl((U'\cap U'')\times V\Bigr)\subseteq\bigcup\mathcal{O}'\cup\bigcup\mathcal{O}''=\bigcup(\mathcal{O}'\cup\mathcal{O}'').$$

Hence $\bigcup\mathcal{W}\in\mathcal{W}_{x}$, which proves $P(n+1)$.

(c) $Y\in\mathcal{W}_{x}$. By (a), $\mathcal{W}_{x}\cap\mathcal{T}_{Y}$ is a subset of $\mathcal{T}_{Y}$ whose union contains $Y$. As $(Y,\mathcal{T}_{Y})$ is compact, Compact Topological Space and Compact Subset §compact gives a finite subset $\mathcal{W}$ of $\mathcal{W}_{x}\cap\mathcal{T}_{Y}$ with $Y\subseteq\bigcup\mathcal{W}$. By Finite Sets §finite there are $n\in\mathbb{N}_{0}$ and a bijection from $[n]$ onto $\mathcal{W}$, so $\bigcup\mathcal{W}\in\mathcal{W}_{x}$ by (b), and therefore $Y\in\mathcal{W}_{x}$ by the closure of $\mathcal{W}_{x}$ under subsets noted above.

Step 2 (finitely many tubes). Let $\mathcal{G}$ be the set of all $U\in\mathcal{T}_{X}$ for which there is a finite subset $\mathcal{O}'$ of $\mathcal{O}$ with $U\times Y\subseteq\bigcup\mathcal{O}'$. For every $x\in X$, Step 1 (c) gives $U\in\mathcal{T}_{X}$ with $x\in U$ and a finite subset $\mathcal{O}'$ of $\mathcal{O}$ with $U\times Y\subseteq\bigcup\mathcal{O}'$, so $x\in U\in\mathcal{G}$. Thus $\mathcal{G}$ is a subset of $\mathcal{T}_{X}$ with $X\subseteq\bigcup\mathcal{G}$, and as $(X,\mathcal{T}_{X})$ is compact, Compact Topological Space and Compact Subset §compact gives a finite subset $\mathcal{G}'$ of $\mathcal{G}$ with $X\subseteq\bigcup\mathcal{G}'$. For $U\in\mathcal{G}'$ let $C_{U}$ be the set of all finite subsets $\mathcal{O}'$ of $\mathcal{O}$ with $U\times Y\subseteq\bigcup\mathcal{O}'$, an element of $\mathcal{P}(\mathcal{P}(\mathcal{O}))$, nonempty because $U\in\mathcal{G}$. Applied to the finite set $\mathcal{G}'$ and the family $(C_{U})_{U\in\mathcal{G}'}$, Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice §choice gives a map $\varphi$ on $\mathcal{G}'$ with $\varphi(U)\in C_{U}$ for every $U\in\mathcal{G}'$. Each $\varphi(U)$ is finite, so the set $\mathcal{O}^{*}=\bigcup_{U\in\mathcal{G}'}\varphi(U)$ is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice §family-union, and it is a subset of $\mathcal{O}$. Let $(x,y)\in X\times Y$. Then $x\in U$ for some $U\in\mathcal{G}'$, and $(x,y)\in U\times Y\subseteq\bigcup\varphi(U)\subseteq\bigcup\mathcal{O}^{*}$. Hence $X\times Y\subseteq\bigcup\mathcal{O}^{*}$.

As $\mathcal{O}$ was arbitrary, $(X\times Y,\mathcal{T})$ is compact by Compact Topological Space and Compact Subset §compact.