Product topology - Imported Knowlpedia Content

The product topology on a product of topological spaces $\{(X_i,\mathcal{T}_i)\}_{i\in I}$ is the topology on the set $\prod_{i\in I}X_i$ generated by the subbasis

\{\pi_i^{-1}(U) : i\in I,\ U\in\mathcal{T}_i\},

where $\pi_i:\prod_{j\in I}X_j\to X_i$ is the $i$ th projection map.

Equivalent characterizations

Equivalently, it is the coarsest topology on $\prod_{i\in I}X_i$ making each projection $\pi_i$ a continuous map.

Remarks

In the common case of two spaces $X\times Y$ , a basis is given by sets of the form $U\times V$ with $U$ open in $X$ and $V$ open in $Y$ .

Examples

$\mathbb{R}^n$ with its usual topology can be viewed as the product of $n$ copies of $\mathbb{R}$ with the usual topology.
If $X$ is discrete and $Y$ is any space, then basic open sets in $X\times Y$ are unions of sets $\{x\}\times V$ with $V$ open in $Y$ .
In an infinite product $\prod_{i\in I}X_i$ , subbasic open sets are “cylinders” that constrain only one coordinate.