Definition:Product Space (Topology)

From ProofWiki
Jump to navigation Jump to search

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 $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$.


Factor Space

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.


$2$ 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 $\struct {S_2, \tau_2}$.


Also known as

The product topology is also known as the Tychonoff topology, named for Andrey Nikolayevich Tychonoff.

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.


Also see

  • Results about product spaces can be found here.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Product_Space_(Topology)&oldid=653044"