nLab
Tychonoff product

Idea

The Tychonoff product (named for Andrey Tikhonov?) is simply the product in the category Top of topological spaces and continuous maps.

It is the usual notion of ‘product’ of topological spaces, but should be distinguished from the box product?, which is sometimes useful. (They are the same for finite products.)

Definitions

Let $(X_i{)}_i$ be a family of topological spaces. Consider the cartesian product ${\prod }_i\mid X_i\mid$ of the underlying sets of these spaces. This set comes with projection map?s ${\pi }_i:{\prod }_j\mid X_j\mid \to X_i$.

Definitions

The Tychonoff topology or product topology on ${\prod }_i\mid X_i\mid$ is the initial topology generated by the ${\pi }_i$. The Tychonoff product or topological product or simply product of the spaces $X_i$ is the set ${\prod }_i\mid X_i\mid$ equipped with the Tychonoff product topology.

Properties

Of course, the maps ${\pi }_i$ are continuous maps, so we have a cone in Top.

Proposition

This is in fact a product cone in $\mathrm{Top}$.