Definition:Tychonoff Topology

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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


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