# Definition:Product Topology/Two Factor Spaces

## 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 $S_2$.

The **product topology $\tau$ on $S_1 \times S_2$** is the topology generated by the natural basis:

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

## 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

- Definition:Product Space (Topology) of Two Factor Spaces: the topological space $\struct {S_1 \times S_2, \tau}$

- Natural Basis of Product Topology of Finite Product which demonstrates the nature of this topology
- Natural Basis of Product Topology
- Product Topology is Coarsest Topology such that Projections are Continuous
- Product Space is Product in Category of Topological Spaces

- Results about
**the product topology**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 $3.5.1$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**product topology**