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Section 2.6 Products

The definition of a product of sets is engineered precisely so that a map into the product is determined by each of its components. More precisely, for any indexed family of sets \(\{X_a\}_{a\in A}\text{,}\) the product

\begin{equation*} \prod_{a\in A}X_a \end{equation*}

is a set equipped with projection maps

\begin{equation*} \pi_b\colon\fromto{\prod_{a\in A}X_a}{X_b}\text{,} \end{equation*}

one for every \(b\in A\text{,}\) which together determine the maps into \(\prod_{a\in A}X_a\text{.}\) That is, composition with the projections induce a bijection

\begin{equation*} \prod_{a\in A}\pi_{a}\circ -\colon\fromto{\Map\left(Y,\prod_{a\in A}X_a\right)}{\prod_{a\in A}\Map(Y,X_a)}\text{.} \end{equation*}

If the sets \(X_a\) are all endowed with topologies, then we would like to topologise the product \(\prod_{a\in A}X_a\) so that the same thing will happen with continuous maps. That is, we would like to ensure that a map \(f\colon\fromto{Y}{\prod_{a\in A}X_a}\) is continuous if and only if, for every \(a\in A\text{,}\) the composite map \(\pi_a\circ f\colon\fromto{Y}{X_a}\) is continuous. But we know just how to arrange that:

Definition 2.6.1.

Let \(\{(X_a,\tau_a)\}_{a\in A}\) be a family of topological spaces. Then the product topology \(\prod_{a\in A}\tau_a\) on \(\prod_{a\in A}X_a\) is the initial topology with respect to the set \(\{\pi_a\}_{\alpha\in A}\text{.}\)

The definition of the product topology is engineered precisely so that a continuous map into the product is determined by each of its continuous components. More precisely, for any indexed family of topological spaces \(\{X_a\}_{a\in A}\text{,}\) the product

\begin{equation*} \prod_{a\in A}X_a \end{equation*}

is a set equipped with continuous projection maps

\begin{equation*} \pi_b\colon\fromto{\prod_{a\in A}X_a}{X_b}\text{,} \end{equation*}

one for every \(b\in A\text{,}\) which together determine the maps into \(\prod_{a\in A}X_a\text{.}\) That is, composition with the projections induce a bijection

\begin{equation*} \prod_{a\in A}\pi_{a}\circ -\colon\fromto{\Map\left(Y,\prod_{a\in A}X_a\right)}{\prod_{a\in A}\Map(Y,X_a)}\text{,} \end{equation*}

In particular, for any set \(A\) and any topological space \(X\text{,}\) we may contemplate the product topology on the set \(\Map(A,X)\text{.}\) We write in particular

\begin{equation*} X^n\coloneq\Map(\{1,\dots,n\},X)\ \ \text{ and } X^{\omega}\coloneq\Map(\NN,X)\text{.} \end{equation*}

The product topology on \(\RR^n\) is generated by sets of the form \(\RR\times\cdots\times \left]a,b\right[\times\RR\times\cdots\times\RR\text{.}\) Consequently, the sets of the form \(C(x,\varepsilon) \coloneq \prod_{i=1}^n\left]x_i-\varepsilon,x_i+\varepsilon\right[\) are a base for the product topology. Now for any \(x \in \RR^n\text{,}\) and for any \(\varepsilon>0\text{,}\) the exists a \(\delta>0\) such that

\begin{equation*} B(x,\delta) \subseteq C(x,\varepsilon)\text{,} \end{equation*}

and

\begin{equation*} C(x,\delta) \subseteq B(x,\varepsilon)\text{.} \end{equation*}

Since the subsets \(C(x, \varepsilon)\) generate the product topology, and the subsets \(B(x,\varepsilon)\) generate the standard topology, the two topologies coincide.

Warning 2.6.4.

Consider the product topological space

\begin{equation*} \RR^{\omega} = \Map(\NN,\RR) = \prod_{n \in \NN} \RR\text{.} \end{equation*}

This topology has a base

\begin{equation*} \left\{ \prod_{i=0}^N \left]a_i, b_i\right[ \times \prod_{i=N+1}^{\infty} \RR \right\}\text{.} \end{equation*}

Consequently, if \(U\) is an open subset of \(\RR^{\omega}\text{,}\) then for any point \(x \in U\text{,}\) there exists \(N \in \NN\) such that any point of the form

\begin{equation*} (x_0,x_1,\dots,x_N,y_{N+1},y_{N+2},\dots) \end{equation*}

also lies in \(U\text{.}\) Conequently, the subset

\begin{equation*} \left]0,1\right[^{\omega} = \Map(\NN, \left]0,1\right[) \subset \RR^{\omega} \end{equation*}

is not open!

If \(S_1,\dots,S_n\) is a finite collection of discrete topological spaces, then the product

\begin{equation*} \prod_{i=1}^n S_i = S_1 \times \cdots \times S_n \end{equation*}

is also discrete. Indeed, for any \(i\) and any \(x_i \in S_i\text{,}\) the set \(S_1 \times \cdots \times S_{i-1} \times \{x_i\} \times S_{i+1} \times \cdots \times S_n\) is open, and since finite intersections of opens are open, it follows that any singleton \(\{(x_1, \dots, x_n)\}\) is open as well.

Warning 2.6.6.

If \(\{S_i\}_{i\in\NN}\) is a countable family of discrete finite sets of cardinality at least \(2\text{,}\) the product \(S \coloneq \prod_{i \in \NN} S_i\) is not discrete: if \((x_0, x_1, \dots) \in S\) is a point, then the open sets are unions of sets of the form

\begin{equation*} \{x_0\} \times \cdots \times \{x_N\} \times \prod_{i=N+1}^{\infty} S_i\text{,} \end{equation*}

so that no singleton is open. In fact, it is always homeomorphic to our old pal the Cantor space \(C\) — irrespective of which finite sets \(S_i\) are chosen!

There is a relative form of the product as well.

Definition 2.6.7.

Let \(U\text{,}\) \(V\text{,}\) and \(X\) be three topological spaces, and let \(f\colon\fromto{U}{X}\) and \(g\colon\fromto{V}{X}\) two continuous maps. Then the fiber product

\(U\times_XV\) is the subspace

\begin{equation*} \left\{(u,v)\in U\times V : f(u)=g(v)\right\} \end{equation*}

of \(U\times V\text{.}\) In this case, the square

\begin{equation*} \end{equation*}

is sometimes called a pullback square.

As a special case of this construction, if \(V\) is a subspace of \(X\) and if \(g\) is the inclusion map \(V \inclusion X\text{,}\) then the fiber product \(U \times_X V\) is homeomorphic to the subspace \(f^{-1}(V) \subseteq X\text{.}\) In particular, if \(V = \{x\}\) for some \(x\in X\text{,}\) then the fiber product \(U \times_X V\) is the fiber \(f^{-1}\{x\}\text{.}\)

Let \(f \colon X \to Y\) be a continuous map of topological spaces. Then the graph of \(f\) is the fiber product

\begin{equation*} \Gamma(f) \coloneq X \operatorname{\times}_{f,Y,\id} Y \subseteq X \times Y\text{.} \end{equation*}

The projection map \(\pr_1 \colon \Gamma(f) \to X\) is a homeomorphism.