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Tychonoff space

In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space is any completely regular space that is also a Hausdorff space; there exist completely regular spaces that are not Tychonoff (i.e. not Hausdorff).

Separation axioms
in topological spaces

(Kolmogorov)

(Fréchet)

(Hausdorff)

(Urysohn)

(completely Hausdorff)

(regular Hausdorff)

(Tychonoff)

(normal Hausdorff)

(completely normal
 Hausdorff)

(perfectly normal
 Hausdorff)

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.[2]

Naming conventions[edit]

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. Some authors, however, switch the meanings of the two kinds of terms, or use all terms interchangeably. 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.

Every is Tychonoff; every pseudometric space is completely regular.

metric space

Every regular space is completely regular, and therefore every locally compact Hausdorff space is Tychonoff.

locally compact

In particular, every is Tychonoff.

topological manifold

Every with the order topology is Tychonoff.

totally ordered set

Every is completely regular.

topological group

Every space is completely regular, but not Tychonoff if the space is not Hausdorff.

pseudometrizable

Every is completely regular (both because it is pseudometrizable and because it is a topological vector space, hence a topological group). But it will not be Tychonoff if the seminorm is not a norm.

seminormed space

Generalizing both the metric spaces and the topological groups, every is completely regular. The converse is also true: every completely regular space is uniformisable.

uniform space

Every is Tychonoff.

CW complex

Every regular space is completely regular, and every normal Hausdorff space is Tychonoff.

normal

The is an example of a Tychonoff space that is not normal.

Niemytzki plane

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. Other examples include:


There are regular Hausdorff spaces that are not completely regular, but such examples are complicated to construct. One of them is the so-called Tychonoff corkscrew,[3][4] which contains two points such that any continuous real-valued function on the space has the same value at these two points. 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.

Properties[edit]

Preservation[edit]

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:

 – a universal map from a topological space X to a compact Hausdorff space βX, such that any map from X to a compact Hausdorff space factors through βX uniquely; if X is Tychonoff, then X is a dense subspace of βX

Stone–Čech compactification