Tychonoff cube

The Tychonoff cube is a universal space for every compact space of weight .

The Hilbert cube, , is a special case of a Tychonoff cube.

The Tychonoff cube is compact.

In fact, one can always choose K to be a Tychonoff cube (i.e. a possibly infinite product of unit intervals).

Every Tychonoff cube is compact Hausdorff as a consequence of Tychonoff's theorem.

A topological space is Tychonoff if and only if it can be embedded in a Tychonoff cube.

Given a cardinal number , we define a Tychonoff cube of weight as the space with the product topology, i.e. the product where is the cardinality of and, for all , .

The Tychonoff cube is named after Andrey Tychonoff, who first considered the arbitrary product of topological spaces and who proved in the 1930s that the Tychonoff cube is compact.

In mathematics, more specifically in general topology, the Tychonoff cube is the generalization of the unit cube from the product of a finite number of unit intervals to the product of an infinite, even uncountable number of unit intervals.