Definition:Product Space (Topology)/General Definition

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Definition

Let $\mathbb S = \left \langle {\left({S_i, \tau_i}\right)}\right \rangle_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.


Let $X$ be the cartesian product of $\mathbb S$:

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


Let $\mathcal T$ be the Tychonoff topology on $S$.


The topological space $\left({S, \mathcal T}\right)$ is called the direct product of $\mathbb S$.


Factor Space

Each of the topological spaces $\left({S_i, \tau_i}\right)$ are called the factors of $\left({S, \mathcal T}\right)$, and can be referred to as factor spaces.


Also see

  • Results about product spaces can be found here.


Sources