Definition:Product Space (Topology)/Two Factor Spaces

Definition
Let $$\mathbb X = \left \langle {\left({X_i, \vartheta_i}\right)}\right \rangle_{i \in I}$$ be a family of topological spaces where $$I$$ is an arbitrary index set.

Let $$X$$ be the cartesian product of $$\mathbb X$$:
 * $$X \ \stackrel {\mathbf {def}} {=\!=} \ \prod_{i \in I} X_i$$

For each $$i \in I$$, let $$\operatorname {pr}_i : X \to X_i$$ be the corresponding projection which maps each ordered tuple in $$X$$ to the corresponding element in $$X_i$$:


 * $$\forall \left({x_i}\right)_{i \in I} \in X: \operatorname {pr}_i \left({\left({x_i}\right)_{i \in I}}\right) = x_i$$

The initial topology $$\mathcal T$$ on $$X$$ with respect to the family $$\left \langle {\operatorname {pr}_i}\right \rangle_{i \in I}$$ is called the product topology on $$X$$.

The topological space $$\left({X, \mathcal T}\right)$$ is called the direct product of the $$\left \langle \left({X_i, \vartheta_i}\right)\right \rangle_{i \in I}$$.

$$\mathcal T$$ is generated by $$\mathcal S = \left\{{ \operatorname {pr}_i^{-1} \left({U}\right) : i \in I, U \in \vartheta_i}\right\}$$.

The basis $$\mathcal S^* = \left\{{\bigcap S : S \subseteq \mathcal S \text{ finite}}\right\}$$ (which is generated by $$\mathcal S$$) is called the natural basis of $$X$$.