Definition:Product Space (Topology)/Two Factor Spaces
This page is about Product Space in the context of Topology. For other uses, see Product Space.
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 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 $\struct {S_2, \tau_2}$.
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.
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
- Natural Basis of Product Topology
- Natural Basis of Product Topology of Finite Product
- Product Topology is Coarsest Topology such that Projections are Continuous
- Product Space is Product in Category of Topological Spaces
- Results about product spaces can be found here.
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.5$: Products: Definition $3.5.1$