Definition:Product Space (Topology)

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Definition

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 Tychonoff topology on $S_1 \times S_2$.

From Natural Basis of Tychonoff Topology of Finite Product, $\tau$ is the topology generated by the basis

$\BB = \set{U_1 \times U_2: U_1 \in \tau_1, U_2 \in \tau_2}$.


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


General 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 $\family {S_i}_{i \mathop \in I}$:

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


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

From Natural Basis of Tychonoff Topology, $\tau$ is generated from:

the basis $\BB$ of cartesian products of the form $\displaystyle \prod_{i \mathop \in I} U_i$ where:
for all $i \in I : U_i \in \tau_i$
for all but finitely many indices $i : U_i = S_i$


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


Factor Space

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


Also known as

The topology on the product space is also known as the product topology and both of the terms product topology and Tychonoff topology are commonly used on $\mathsf{Pr} \infty \mathsf{fWiki}$.


Other sources may use the terms direct product, topological product or Tychonoff product for the product space, but these terms are less precise and there exists the danger of confusion with other similar uses of these terms in different contexts.


Also see

  • Results about product spaces can be found here.


Sources