Definition:Product Space (Topology)

This page is about product space in the context of topology. For other uses, see Product Space.

Definition

Let $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

Let $S$ be the cartesian product of $\family {S_i}_{i \mathop \in I}$:

$\ds S := \prod_{i \mathop \in I} S_i$

Let $\tau$ be the product topology on $S$.

The topological space $\struct {X, \tau}$ is called the product space of $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$.

Each of the topological spaces $\struct {X_i, \tau_i}$ are called the factors of $\struct {\XX, \tau}$, and can be referred to as factor spaces.

Let $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$ be topological spaces.

Let $S_1 \times S_2$ be the cartesian product of $S_1$ and $S_2$.

Let $\tau$ be the product topology on $S_1 \times S_2$.

The topological space $\struct {S_1 \times S_2, \tau}$ is called the product space of $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$.

While both of these terms are commonly used on $\mathsf{Pr} \infty \mathsf{fWiki}$, the preference is for product topology.

Various other terms can be found in the literature for the product space, for example:

direct product

topological product

Tychonoff product

but these terms are less precise, and there exists the danger of confusion with other similar uses of these terms in different contexts.

Note that the topological space $\struct {\XX, \tau}$ itself is never referred to as a Tychonoff space.

This is because a Tychonoff space is a different concept altogether.