Definition:Product Space (Topology)
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Definition
Let $T_1 = \left({X_1, \vartheta_1}\right)$ and $T_2 = \left({X_2, \vartheta_2}\right)$ be topological spaces.
Let $X_1 \times X_2$ be the cartesian product of $X_1$ and $X_2$.
The product topology $\vartheta$ for $X_1 \times X_2$ is the topology with basis $\mathcal B = \left\{{U_1 \times U_2: U_1 \in \vartheta_1, U_2 \in \vartheta_2}\right\}$.
Indicate how the product topology is the same as the Tychonoff topology.
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General 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 $\mathbb X$:
- $\displaystyle X := \prod_{i \in I} X_i$
Let $\mathcal T$ be the Tychonoff topology on $X$.
The topological space $\left({X, \mathcal T}\right)$ is called the direct product of $\mathbb X$.
Factor Space
Each of the topological spaces $\left({X_i, \vartheta_i}\right)$ are called the factors of $\left({X, \mathcal T}\right)$, and can be referred to as factor spaces.
Also see
- Results about product spaces can be found here.
Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Functions
