Tychonoff space

Type of regular Hausdorff 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).

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]

Definitions

Separation of a point from a closed set via a continuous function.

A topological space ${\displaystyle X}$ 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 ${\displaystyle A\subseteq X}$ and any point ${\displaystyle x\in X\setminus A,}$ there exists a real-valued continuous function ${\displaystyle f:X\to \mathbb {R} }$ such that ${\displaystyle f(x)=1}$ and ${\displaystyle f\vert _{A}=0.}$ (Equivalently one can choose any two values instead of ${\displaystyle 0}$ and ${\displaystyle 1}$ and even require that ${\displaystyle f}$ be a bounded function.)

A topological space is called a Tychonoff space (alternatively: T_3½ space, or T_π space, or completely T₃ space) if it is a completely regular Hausdorff space.

Remark. Completely regular spaces and Tychonoff spaces are related through the notion of Kolmogorov equivalence. A topological space is Tychonoff if and only if it's both completely regular and T0. On the other hand, a space is completely regular if and only if its Kolmogorov quotient is Tychonoff.

Naming conventions

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.

Examples

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:

• Every metric space is Tychonoff; every pseudometric space is completely regular.
• Every locally compact regular space is completely regular, and therefore every locally compact Hausdorff space is Tychonoff.
• In particular, every topological manifold is Tychonoff.
• Every totally ordered set with the order topology is Tychonoff.
• Every topological group is completely regular.
• Every pseudometrizable space is completely regular, but not Tychonoff if the space is not Hausdorff.
• Every seminormed space 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.
• Generalizing both the metric spaces and the topological groups, every uniform space is completely regular. The converse is also true: every completely regular space is uniformisable.
• Every CW complex is Tychonoff.
• Every normal regular space is completely regular, and every normal Hausdorff space is Tychonoff.
• The Niemytzki plane is an example of a Tychonoff space that is not normal.

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

Preservation

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:

• Every subspace of a completely regular or Tychonoff space has the same property.
• A nonempty product space is completely regular (respectively Tychonoff) if and only if each factor space is completely regular (respectively Tychonoff).

Like all separation axioms, complete regularity is not preserved by taking final topologies. In particular, quotients of completely regular spaces need not be regular. 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.

Real-valued continuous functions

For any topological space ${\displaystyle X,}$ let ${\displaystyle C(X)}$ denote the family of real-valued continuous functions on ${\displaystyle X}$ and let ${\displaystyle C_{b}(X)}$ be the subset of bounded real-valued continuous functions.

Completely regular spaces can be characterized by the fact that their topology is completely determined by ${\displaystyle C(X)}$ or ${\displaystyle C_{b}(X).}$ In particular:

• A space ${\displaystyle X}$ is completely regular if and only if it has the initial topology induced by ${\displaystyle C(X)}$ or ${\displaystyle C_{b}(X).}$
• A space ${\displaystyle X}$ is completely regular if and only if every closed set can be written as the intersection of a family of zero sets in ${\displaystyle X}$ (i.e. the zero sets form a basis for the closed sets of ${\displaystyle X}$).
• A space ${\displaystyle X}$ is completely regular if and only if the cozero sets of ${\displaystyle X}$ form a basis for the topology of ${\displaystyle X.}$

Given an arbitrary topological space ${\displaystyle (X,\tau )}$ there is a universal way of associating a completely regular space with ${\displaystyle (X,\tau ).}$ Let ρ be the initial topology on ${\displaystyle X}$ induced by ${\displaystyle C_{\tau }(X)}$