The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space

The product set. Let $I$ be a set and let $X_i$ be a set for each $i\in I$. The product is

${\prod }_{i\in I}{X}_i:=\{x:x\text{ is a function with domain }I\text{ and }x(i)\in X_i\text{ for every }i\in I\},$

and we write $x_i:=x(i)$, the $i$-th coordinate of $x$. Two elements of the product are equal exactly when they agree at every index, functions being equal when they have the same domain and the same values. For $j\in I$ the $j$-th projection is

${\pi }_j:{\prod }_{i\in I}{X}_i\to X_j,{\pi }_j(x):=x_j\mathrm{.}$

Notation for a finite product. For $I=n$ a natural number, which is the set $\{0,1,\dots ,n-1\}$ of its predecessors, an element of ${\prod }_{k<n}{X}_k$ is a function on $n$ and we write it $(x_0,\dots ,x_{n-1})$. In particular $I=2$ gives the binary product, written $X\times Y$ for ${\prod }_{i<2}{X}_i$ with $X_0=X$ and $X_1=Y$, whose elements are written $(u,v)$ for the function $0\mapsto u$, $1\mapsto v$. This is the only meaning the symbol $X\times Y$ carries on this page.

Two facts about when the product is nonempty, stated because they are used and because they cost something. If some $X_{i_0}$ is empty then the product is empty, since no function can take a value in $X_{i_0}$. Conversely, suppose every $X_i$ is nonempty.

• For $I=n$ a natural number, the product is nonempty, and this is a theorem of ZF: Every natural-number-indexed list of nonempty sets has a choice function on its family of values applied to the function $i\mapsto X_i$ on $n$ supplies a choice function $g$ for the family of values, and $x(i):=g(X_i)$ defines a member of ${\prod }_{k<n}{X}_k$.
• For an arbitrary $I$ the assertion "${\prod }_{i\in I}{X}_i\mathrm{\ne }\mathrm{\varnothing }$ whenever every $X_i$ is nonempty" is the Axiom of Choice: it is the formulation recorded in The Axiom of Choice, and the choice function of Choice function is exactly a point of the product of a family by itself. Every use of it below is flagged at the step that spends it.

The box topology. Now let each $X_i$ carry a topology ${\mathcal{T}}_i$ (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison). Put

$\mathcal{R}:=\{{\prod }_{i\in I}{U}_i:U_i\in {\mathcal{T}}_i\text{ for every }i\in I\},$

the family of boxes. $\mathcal{R}$ is a basis for a topology (A family is a basis for a unique topology iff it covers the set and every point of an intersection of two members lies in a member inside that intersection; finite intersections of any subbasis form a basis): it contains ${\prod }_i{X}_i$, so it covers the product, and it is closed under binary intersections, since

$\left({\prod }_i{U}_i\right)\cap \left({\prod }_i{V}_i\right)={\prod }_i(U_i\cap V_i)$

and each $U_i\cap V_i$ is open by (T3). The topology it generates is the box topology ${\mathcal{T}}^{\mathrm{\square }}$, and $\mathcal{R}$ is a basis for it (Basis and subbasis for a topology, and the topology generated by a family of sets).

The product topology. The product topology ${\mathcal{T}}^{\mathrm{\Pi }}$ on ${\prod }_i{X}_i$ is the initial topology of the family of projections $({\pi }_i{)}_{i\in I}$ (The initial topology of a family of maps into spaces and the final topology of a family of maps out of spaces, and the subspace topology as the model initial topology): the topology generated by the subbasis

$\mathcal{G}:=\{{\pi }_i^{-1}[U]:i\in I,U\in {\mathcal{T}}_i\},{\pi }_i^{-1}[U]={\prod }_{j\in I}{W}_j\text{ with }{W}_i=U\text{ and }{W}_j=X_j\text{ for }j\mathrm{\ne }i\mathrm{.}$

By A family is a basis for a unique topology iff it covers the set and every point of an intersection of two members lies in a member inside that intersection; finite intersections of any subbasis form a basis the finite intersections of members of $\mathcal{G}$ form a basis for ${\mathcal{T}}^{\mathrm{\Pi }}$, and those finite intersections are exactly the boxes with all but finitely many factors unrestricted:

${\mathcal{R}}^{\mathrm{\Pi }}=\{{\prod }_{i\in I}{U}_i:U_i\in {\mathcal{T}}_i\text{ for every }i,\text{ and }{U}_i=X_i\text{ for all but finitely many }i\}\mathrm{.}$

Indeed the intersection of ${\pi }_{i_1}^{-1}[U_1],\dots ,{\pi }_{i_n}^{-1}[U_n]$ is the box whose factor at $i$ is the intersection of those $U_m$ with $i_m=i$ and is $X_i$ when no $i_m$ equals $i$; and the intersection of no members is the whole product, the box with every factor $X_i$. Conversely a box with $U_i=X_i$ off a finite set is such an intersection. Members of ${\mathcal{R}}^{\mathrm{\Pi }}$ are called basic product-open sets, and members of $\mathcal{R}$ boxes. So ${\mathcal{R}}^{\mathrm{\Pi }}\subseteq \mathcal{R}$, with equality when $I$ is a natural number.

The empty product. For $I=\mathrm{\varnothing }$ there is exactly one function with domain $\mathrm{\varnothing }$, the empty function, so ${\prod }_{i\in \mathrm{\varnothing }}{X}_i$ is a one-point set. A one-point set carries exactly one topology, namely $\{\mathrm{\varnothing },\{\mathrm{\varnothing }\}\}$, since a topology must contain the empty set and the whole set and there is nothing else to contain (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison); so the box topology and the product topology agree there, and both equal the discrete topology and the indiscrete topology (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies), which coincide on a one-point set. There are no projections to speak of, and the initial topology of the empty family is indeed the indiscrete one (The initial topology of a family of maps into spaces and the final topology of a family of maps out of spaces, and the subspace topology as the model initial topology).

Convention. Unless the box topology is named explicitly, ${\prod }_i{X}_i$ always carries the product topology in this library. That is not a matter of taste: the product topology is the one with the characteristic property of the next item, and the box topology has no such property.

• Where the two topologies actually differ. The box topology is finer than the product topology by construction, since ${\mathcal{R}}^{\mathrm{\Pi }}\subseteq \mathcal{R}$. They agree whenever $I$ is finite; and, assuming the Axiom of Choice, for a family of nonempty spaces they differ for infinite $I$ as soon as infinitely many factors have a nonempty proper open subset. Nonemptiness is not decoration: if one factor is empty then the product is empty and carries exactly one topology, so the two agree however the other factors are chosen. Both statements are proved two items below, with that hypothesis, and the failure is recorded on this page as a false statement.

• The product set is a set of functions, and that is not a technicality. The factors are indexed by an arbitrary set, so there is no "list" to write down; writing $x=(x_i{)}_{i\in I}$ is notation for the function $x$. The finite case recovers the familiar tuple, and the identification of ${\prod }_{k<n}\mathbb{R}$ with the ${\mathbb{R}}^n$ of ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it is literal, that item defining ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$.

• The projections carry no hypothesis. They are defined for every product, including the empty one and products with an empty factor; what does need a hypothesis is their surjectivity, which is the point at which choice enters and which is stated separately in the next item.