Tychonoff space

Paul Urysohn had used the notion of completely regular space in a 1925 paper[1] without giving it a name.

But it was Andrey Tychonoff who introduced the terminology completely regular in 1930.

is called completely regular if points can be separated from closed sets via (bounded) continuous real-valued functions.

In technical terms this means: for any closed set

On the other hand, a space is completely regular if and only if its Kolmogorov quotient is Tychonoff.

Across mathematical literature different conventions are applied when it comes to the term "completely regular" and the "T"-Axioms.

The definitions in this section are in typical modern usage.

In Wikipedia, the terms "completely regular" and "Tychonoff" are used freely and the "T"-notation is generally avoided.

In standard literature, caution is thus advised, to find out which definitions the author is using.

For more on this issue, see History of the separation axioms.

Almost every topological space studied in mathematical analysis is Tychonoff, or at least completely regular.

For example, the real line is Tychonoff under the standard Euclidean topology.

An even more complicated construction starts with the Tychonoff corkscrew and builds a regular Hausdorff space called Hewitt's condensed corkscrew,[5][6] which is not completely regular in a stronger way, namely, every continuous real-valued function on the space is constant.

Complete regularity and the Tychonoff property are well-behaved with respect to initial topologies.

Specifically, complete regularity is preserved by taking arbitrary initial topologies and the Tychonoff property is preserved by taking point-separating initial topologies.

It follows that: Like all separation axioms, complete regularity is not preserved by taking final topologies.

Quotients of Tychonoff spaces need not even be Hausdorff, with one elementary counterexample being the line with two origins.

There are closed quotients of the Moore plane that provide counterexamples.

denote the family of real-valued continuous functions on

or, equivalently, the topology generated by the basis of cozero sets in

This construction is universal in the sense that any continuous function

In the language of category theory, the functor that sends

is left adjoint to the inclusion functor CReg → Top.

By taking Kolmogorov quotients, one sees that the subcategory of Tychonoff spaces is also reflective.

The algebraic theory of these rings is therefore subject of intensive studies.

A vast generalization of this class of rings that still resembles many properties of Tychonoff spaces, but is also applicable in real algebraic geometry, is the class of real closed rings.

to be a Tychonoff cube (i.e. a possibly infinite product of unit intervals).

Since every subspace of a compact Hausdorff space is Tychonoff one has: Of particular interest are those embeddings where the image of

Complete regularity is exactly the condition necessary for the existence of uniform structures on a topological space.

A topological space admits a separated uniform structure if and only if it is Tychonoff.