# Definition:Tychonoff Topology

## Contents

## Definition

Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$:

- $\displaystyle X := \prod_{i \mathop \in I} X_i$

For each $i \in I$, let $\pr_i: X \to X_i$ denote the $i$th projection on $X$:

- $\forall \family {x_j}_{j \mathop \in I} \in X: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$

The **Tychonoff topology** on $X$ is defined as the initial topology $\tau$ on $X$ with respect to $\family {\pr_i}_{i \mathop \in I}$.

By definition of the initial topology on $X$ with respect to $\family {\pr_i}_{i \mathop \in I}$, $\tau$ is generated by the sub-basis:

- $\SS = \set {\pr_i^{-1} \sqbrk U: i \in I, \, U \in \tau_i}$

### Natural Sub-Basis

The **natural sub-basis on $X$** is defined as:

- $\mathcal S = \set {\pr_i^{-1} \sqbrk U: i \in I, \, U \in \tau_i}$

### Natural Basis

The **natural basis on $X$** is defined as the basis generated by $\SS$.

## Also known as

The **Tychonoff topology** is also known as the **product topology** and both of these terms are commonly used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also see

- Natural Basis of Tychonoff Topology
- Natural Basis of Tychonoff Topology of Finite Product
- Tychonoff Topology is Coarsest Topology such that Projections are Continuous
- Product Space is Product in Category of Topological Spaces
- Definition:Product Space (Topology)
- Results about
**Tychonoff topologies**can be found here.

### Relation between Tychonoff and Box Topology

- Results about the relation between the
**Tychonoff topology**and the box topology can be found here.

## Source of Name

This entry was named for Andrey Nikolayevich Tychonoff.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 1$: Functions