Product topology - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

The product topology on a family of topological spaces $\{(X_i,\mathcal T_i)\}_{i\in I}$ is the topology on $\prod_{i\in I}X_i$ 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.

Equivalent characterizations

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

Remarks

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

Examples

The usual topology on $\mathbb R^n$ is the product topology of $n$ copies of $\mathbb R$ .
If $X$ is discrete and $Y$ is any space, the sets $\{x\}\times V$ , with $V$ open in $Y$ , form a basis for $X\times Y$ .
In an infinite product $\prod_{i\in I}X_i$ , a basic open set constrains only finitely many coordinates.