Tychonoff's Theorem

Theorem
Let $\left \langle {X_i}\right \rangle_{i \in I}$ be a family of non-empty topological spaces, where $I$ is an arbitrary index set.

Let $\displaystyle X = \prod_{i \mathop \in I} X_i$ be the corresponding product space.

Then $X$ is compact if and only if each $X_i$ is.

Proof
First assume that $X$ is compact.

Since the projections $\operatorname{pr}_i : X \to X_i$ are continuous, it follows that the $X_i$ are compact.

Assume now that each $X_i$ is compact.

By the equivalent definitions of compact sets it is enough to show that every ultrafilter on $X$ converges.

Thus let $\mathcal F$ be an ultrafilter on $X$.

For each $i \in I$, the image filter $\operatorname{pr}_i \left({\mathcal F}\right)$ then is an ultrafilter on $X_i$.

Since each $X_i$ is compact by assumption, each $\operatorname{pr}_i \left({\mathcal F}\right)$ therefore converges to a $x_i \in X_i$.

This implies that $\mathcal F$ converges to $x := \left({x_i}\right)_{i \in I}$.

Preliminaries
From the definition of the Tychonoff topology, a basic open set of the natural basis of $X$ is a set of the form:
 * $\displaystyle \prod_{i \mathop \in I} U_i$

where:
 * each $U_i$ is a nonempty open subset of $X_i$

and:
 * $U_i = X_i$ for all but finitely many $i \in I$.

The word collection will be used to mean set of sets.

Mappings of initial intervals of a well-ordered set form a set-theoretic tree if ordered by inclusion, see Mappings of Initial Intervals of a Well-Ordered Set Ordered by Inclusion.

Statement that holds in Zermelo–Fraenkel set theory (without Axiom of Choice)
Let $\left({I, <}\right)$ be a well-ordered set.

Let $\left \langle {X_i} \right \rangle_{i \in I}$ be a family of compact topological spaces.

Denote $\displaystyle X = \prod_{i \mathop \in I} X_i$.

Let $\left({F, \subset}\right)$ be the set-theoretic tree:
 * $\displaystyle \left({\bigcup_{i \mathop \in I} \prod_{j \mathop < i} X_j}\right) \cup X$

of mappings defined on initial intervals of $I$, where the ordering is that of set inclusion.

Consider the set of all subtrees $T \subseteq F$ with the following property:
 * For every $i \in I$ and every $\displaystyle f \in T \cap \prod_{j \mathop < i} X_j$, the set $\left\{{g(i): g \in T,\ f \subsetneq g}\right\}$ is closed in $X_i$.

Suppose that every such subtree $T$ of $F$ has a branch.

Then $\displaystyle \prod_{i \mathop \in I} X_i$ is compact.

Proof of the version without Axiom of Choice
To prove that every open cover of $X$ has a finite subcover, it is enough to prove that every open cover by basic open set of the natural basis has a finite subcover.

Let $\mathcal O$ be a collection of basic open subsets of $X$ such that no finite subcollection of $\mathcal O$ covers $X$.

It is enough to prove that $\mathcal O$ does not cover $X$.

Let $T_{\mathcal O}$ be the set of all elements $f \in F$ such that the set $\left\{{x \in X : f \subseteq x}\right\}$ is not covered by any finite subcollection of $\mathcal O$.

Then $T_{\mathcal O}$ is a subtree of $F$.

For every $i \in I$ and $\displaystyle f \in T_{\mathcal O} \cap \prod_{j \mathop < i} X_j$, denote:


 * $C_{\mathcal O} \left({f}\right) = \left\{{g \left({i}\right): f \subsetneq g \in T_{\mathcal O}}\right\}$

Step 1.
For every $i \in I$ and every $\displaystyle f \in T_{\mathcal O} \cap \prod_{j \mathop < i} X_j$, $C_{\mathcal O} \left({f}\right)$ is closed in $X_i$ and nonempty.

To prove this, let $\mathcal W$ be the collection of all open subsets $U$ of $X_i$ such that there exist a finite subcollection $\mathcal P \subseteq \mathcal O$ such that:
 * $\displaystyle \left\{{x \in X: f \subseteq x,\ x \left({i}\right) \in U}\right\}\subseteq \bigcup \mathcal P$

Then $\displaystyle X_i \setminus C_{\mathcal O} \left({f}\right) = \bigcup \mathcal W$ and hence $C_{\mathcal O} \left({f}\right)$ is closed.

Suppose $C_{\mathcal O} \left({f}\right)$ is empty.

Then $\mathcal W$ would be a cover for $X_i$ and, by compactness of $X_i$, it would have a finite subcover.

This would yield a finite subcollection of $\mathcal O$ that covers $\left\{{x \in X: f \subseteq x}\right\}$ in contradiction with the fact that $f \in T_{\mathcal O}$.

So $C_{\mathcal O} \left({f}\right)$ is nonempty.

It is left to show that indeed $\displaystyle X_i \setminus C_{\mathcal O} \left({f}\right) = \bigcup \mathcal W$.

Consider an arbitrary $a \in X_i \setminus C_{\mathcal O} \left({f}\right)$ and define $\displaystyle g \in \prod_{j \mathop \le i} X_j$ by: $f \subseteq g$ and $g \left({i}\right) = a$.

Then $g \notin T_{\mathcal O}$, and therefore there is a finite collection $\mathcal P \subseteq \mathcal O$ such that:


 * $\displaystyle \left\{{x \in X: f \subseteq x, x \left({i}\right) = a}\right\} = \left\{{x \in X: g \subseteq x}\right\} \subseteq \bigcup \mathcal P$

and
 * $\left\{{x \in X: f \subseteq x, x \left({i}\right) = a}\right\} \cap V \ne \varnothing$ for every $V \in \mathcal P$.

Let $\displaystyle U = \bigcap \left\{{\operatorname{pr}_i \left({V}\right): V \in \mathcal P}\right\}$.

Then $U$ is an open subset of $X_i$, $a\in U$, and


 * $\displaystyle \left\{{x \in X: f \subseteq x, x \left({i}\right) \in U}\right\} \subseteq \bigcup \mathcal P$

Therefore $U \cap C_{\mathcal O} \left({f}\right) = \varnothing$ and $a \in U \in \mathcal W$.

Step 2.
Every branch of $T_{\mathcal O}$ has the greatest element.

To prove this, suppose that $B$ is a branch of $T_{\mathcal O}$ without the greatest element.

Let $f = \bigcup B$.

Let $i$ be the least element of $I$ that is not in the domain of any element of $B$.

Then $\displaystyle f \in \prod_{j \mathop < i} X_j$.

Since $B$ has no greatest element, $f \notin B$, and since $B$ is a maximal chain in $T_{\mathcal O}$, it follows that $f \notin T_{\mathcal O}$.

Let $\mathcal P \subseteq \mathcal O$ be a finite collection such that:
 * $\displaystyle \left\{{x \in X: f \subseteq x}\right\} \subseteq \bigcup \mathcal P$

Let $m$ be the greatest element of the finite set:
 * $\left\{{j \in I: j < i \text{ and } \exists V \in \mathcal P: \left({\operatorname{pr}_j \left({V}\right) \ne X_j}\right)}\right\}$

Let $g$ be any element of $B$ that is defined on $m$.

Consider an arbitrary $x \in X$ such that $g \subseteq x$.

Let $y \in X$ be defined by $f \subseteq y$ and $y \left({j}\right) = x \left({j}\right)$ for every $j \ge i$, and choose $V \in \mathcal P$ such that $y \in V$.

Then $x \in V$ (because $V$ does not "take into account" the values of $x \left({j}\right)$ for $m < j < i$).

Thus:
 * $\left\{{x \in X: g \subseteq x}\right\} \subseteq \bigcup \mathcal P$

in contradiction of the fact that $g \in T_{\mathcal O}$.

Step 3.
Every maximal element of $T_{\mathcal O}$ is an element of $X$: an element $f \in T_{\mathcal O}\setminus X$ cannot be maximal in $T_{\mathcal O}$ because $C_{\mathcal O} \left({f}\right) \ne \varnothing$.

Conclusion.
Now it can be shown that there is $f \in X$ such that $f \notin \bigcup \mathcal O$.

Indeed, according to the hypotheses, $T_{\mathcal O}$ has a branch $B$.

Let $f$ be the greatest element of $B$.

Then $f$ is a maximal element of $T_{\mathcal O}$.

Therefore $f \in X$.

Therefore the set $\left\{{f}\right\} = \left\{{x \in X: f \subseteq x}\right\}$ is not covered by any finite subcollection of $\mathcal O$.

Hence $f \notin \bigcup \mathcal O$.

Corollary 1
The Cartesian product of a finite family of compact topological spaces is compact.

(This is also proved in Topological Product of Compact Spaces.)

Corollary 2
Let $I$ be a well-orderable set and $\left \langle {X_i}\right \rangle_{i \in I}$ a family of compact topological spaces.

Suppose that the Cartesian product of all nonempty closed subsets of all $X_i$ is nonempty:
 * $\displaystyle \prod \left\{{C: C \ne \varnothing \text { and } C \text{ is closed in } X_i \text { for some } i \in I}\right\} \ne \varnothing$

Then $\displaystyle \prod_{i \mathop \in I} X_i$ is compact.

Proof
Let $<$ be a well-order relation on $I$.

Denote $\displaystyle X = \prod_{i \mathop \in I} X_i$.

Let $F$ be the tree $\displaystyle \left({\bigcup_{i \mathop \in I} \prod_{j \mathop < i} X_j}\right) \cup X$ ordered by set inclusion.

To apply the theorem, it is enough to verify that:
 * If $T$ is a subtree of $F$ such that for every $i \in I$ and every $\displaystyle f \in T \cap \prod_{j \mathop < i} X_j$, the set $\left\{{g \left({i}\right): g \in T,\ f \subsetneq g}\right\}$ is closed in $X_i$, then $T$ has a branch.

Let:
 * $\displaystyle e \in \prod \left\{{C: C \ne \varnothing \text { and } C \text{ is closed in } X_i \text { for some } i \in I}\right\}$

be a choice function.

That is $e(C)\in C$ for every nonempty $C$ which is closed in some $X_i$.

Let $B_e$ be the minimal tree among all subtrees $S$ of $T$ with the property that for every $i \in I$ and every $\displaystyle f \in S \cap \prod_{j \mathop < i} X_j$:


 * $e \left({\left\{{g \left({i}\right): g \in T,\ f \subsetneq g}\right\}}\right) \in \left\{{g \left({i}\right): g \in S,\ f \subsetneq g}\right\}$

unless
 * $\left\{{g \left({i}\right): g \in T,\ f \subsetneq g}\right\} = \varnothing$

Such subtrees of $T$ exist because $T$ itself is such, and the minimal such subtree is the intersection of all such subtrees.

It can be shown that $B_e$ is a branch by assuming that it is not, considering the minimal $i\in I$ where it "branches", and arriving at a contradiction with its minimality.

Applications
The proof that $\left[{0 \,.\,.\, 1}\right]^\Z$ is compact does not require the Axiom of Choice, because the product of all nonempty closed subsets of $\left[{0 \,.\,.\, 1}\right]$ contains, for example, the greatest lower bound function $\inf$ (restricted to the collection of closed subsets of $\left[{0 \,.\,.\, 1}\right]$):


 * $\displaystyle \left\{{(C,\inf C) : C \ne \varnothing \text { and } C \text{ is closed in } \left[{0 \,.\,.\, 1}\right]}\right\}

\in\prod \left\{{C: C \ne \varnothing \text { and } C \text{ is closed in } \left[{0 \,.\,.\, 1}\right]}\right\}$

(Here $\displaystyle \left\{{(C,\inf C) : C \ne \varnothing \text { and } C \text{ is closed in } \left[{0 \,.\,.\, 1}\right]}\right\}$ is a way to write a function $f$ defined on the collection of all nonempty closed subsets of $[0 \,.\,.\, 1]$ by $f(C)=\inf C$.)