Product Topology

The product topology is the standard topology on a Cartesian product of topological spaces. It is the coarsest topology—that is, the one with the fewest open sets—for which every coordinate projection is a continuous function. For infinitely many factors, its defining feature is that a basic open set restricts only finitely many coordinates, leaving all others unrestricted. This construction makes continuity and convergence into a product equivalent to their coordinatewise counterparts. (people.math.osu.edu)

Definition and basic open sets

Let $\{X_i\}_{i\in I}$ be a family of topological spaces indexed by a set $I$, and write

A point of $X$ is a tuple $x=(x_i)_{i\in I}$, with $x_i\in X_i$. The coordinate projection is the function

The product topology has a basis consisting of sets

where each $U_i$ is an open set in $X_i$, and $U_i=X_i$ for all but finitely many indices. Its open sets are arbitrary unions of these basic sets. Equivalently, the sets $\pi_i^{-1}(U_i)$ form a subbasis: finite intersections of them generate the basis. (math.ucla.edu)

Thus a basic neighborhood of $x$ specifies open neighborhoods of $x_i$ in a finite collection of factors. Different neighborhoods can restrict different coordinates; there need not be one finite collection controlling the entire topology. For finite products, the restriction is automatic, so the usual open rectangles form a basis. In particular, the product of $n$ copies of the real line has the ordinary topology of Euclidean space $\mathbb R^n$. (math.wustl.edu)

Universal property and continuity

The product topology satisfies a universal property. Given a topological space $Y$ and continuous maps $f_i:Y\to X_i$, there is a unique continuous map

such that $\pi_i\circ f=f_i$ for every $i$. Consequently, an arbitrary map $f:Y\to X$ is continuous if and only if all its coordinate functions are continuous. In category theory, this says that $X$, together with its projections, is the product of the spaces $X_i$ in the category of topological spaces and continuous maps. (math.ucla.edu)

The converse direction of the continuity criterion follows because inverse images of basic open sets are finite intersections of open sets:

Coordinate projections are also open maps. This criterion concerns maps into a product; it does not provide an analogous automatic test for maps out of one. (math.ucla.edu)

Coordinatewise convergence

A net $x_\alpha$ converges to $x$ in the product topology exactly when

Necessity follows from continuity of the projections. For sufficiency, a basic neighborhood imposes finitely many conditions, and coordinatewise convergence ensures that all are eventually satisfied simultaneously. The same equivalence holds for sequences. (people.math.osu.edu)

If every factor is the same space $Z$, the product $Z^I$ can be identified with the set of functions $I\to Z$. Product convergence then becomes pointwise convergence, explaining the alternative name topology of pointwise convergence. It requires convergence at each individual input, rather than a common bound on errors across all inputs, as in uniform convergence. (people.math.osu.edu)

Comparison with the box topology

The box topology instead allows every product $\prod_i U_i$ of open sets as a basic open set, without requiring almost all $U_i$ to equal $X_i$. It therefore contains the product topology. The two agree for finite products but can differ for infinite products. For example,

is box-open in $\mathbb R^{\mathbb N}$, but not product-open: any product-basic neighborhood of the zero tuple leaves some coordinate unrestricted. (math.ucla.edu)

This distinction affects continuity. The diagonal map $t\mapsto(t,t,\ldots)$ from $\mathbb R$ into the product is continuous because each coordinate is continuous. Into the box topology it is not: the inverse image of the displayed box is $\{0\}$, which is not open in $\mathbb R$. Thus unrestricted coordinate conditions destroy the product’s universal continuity criterion. (legacy-www.math.harvard.edu)

Compactness, connectedness, and examples

Tychonoff’s theorem states that an arbitrary product of compact spaces is compact in the product topology. Standard proofs use the axiom of choice, for example through Alexander’s subbasis theorem. Products also preserve connectedness: a product of connected spaces is connected. (math.ucla.edu)

Two characteristic examples are $[0,1]^{\mathbb N}$, the Hilbert cube, and $\{0,1\}^{\mathbb N}$, with each two-point factor discrete. Both are compact. The latter is homeomorphic to the Cantor set; its basic neighborhoods prescribe finitely many binary coordinates. (legacy-www.math.harvard.edu)

Metrizability

A countable product of metric spaces $(X_n,d_n)$ is metrizable. One compatible metric is

The summable weights make the unrestricted tail uniformly small, while finitely many initial coordinates control the remaining distance. (math.wustl.edu)

Countability matters: an uncountable product of metric spaces, each containing at least two points, is not a first-countable space and hence is not metrizable. A countable collection of basic neighborhoods restricts only countably many coordinates altogether, leaving another coordinate available to impose a neighborhood condition that the collection cannot refine. (math.wustl.edu)