Intopology and related branches ofmathematics,Tychonoff spaces andcompletely regular spaces are kinds oftopological spaces. These conditions are examples ofseparation axioms. A Tychonoff space is any completely regular space that is also aHausdorff 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 wasAndrey Tychonoff who introduced the terminologycompletely regular in 1930.^[2]

A topological space${\displaystyle X}$ is calledcompletely regular if points can beseparated from closed sets via (bounded) continuous real-valued functions. In technical terms this means: for anyclosed set${\displaystyle A\subseteq X}$ and anypoint${\displaystyle x\in X\setminus A,}$ there exists areal-valuedcontinuous function${\displaystyle f:X\to \mathbb{R}}$ such that${\displaystyle f(x)=1}$ and${\displaystyle f{|}_A=0.}$ (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 aTychonoff space (alternatively:T_3½ space, orT_π space, orcompletely T₃ space) if it is a completely regularHausdorff space.

Remark. Completely regular spaces and Tychonoff spaces are related through the notion ofKolmogorov equivalence. A topological space is Tychonoff if and only if it's both completely regular andT₀. On the other hand, a space is completely regular if and only if itsKolmogorov 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. 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, seeHistory of the separation axioms.

Almost every topological space studied inmathematical analysis is Tychonoff, or at least completely regular.For example, thereal line is Tychonoff under the standardEuclidean topology.Other examples include:

• Everymetric space is Tychonoff; everypseudometric space is completely regular.
• Everylocally compactregular space is completely regular, and therefore every locally compact Hausdorff space is Tychonoff.
• In particular, everytopological manifold is Tychonoff.
• Everytotally ordered set with theorder topology is Tychonoff.
• Everytopological group is completely regular.
• Everypseudometrizable space is completely regular, but not Tychonoff if the space is not Hausdorff.
• Everyseminormed space is completely regular (both because it is pseudometrizable and because it is atopological 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, everyuniform space is completely regular. The converse is also true: every completely regular space is uniformisable.
• EveryCW complex is Tychonoff.
• Everynormal regular space is completely regular, and every normal Hausdorff space is Tychonoff.
• TheNiemytzki plane is an example of a Tychonoff space that is notnormal.

There are regular Hausdorff spaces that are not completely regular, but such examples are complicated to construct. One of them is the so-calledTychonoff 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 calledHewitt'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 toinitial 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:

• Everysubspace of a completely regular or Tychonoff space has the same property.
• A nonemptyproduct 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 takingfinal topologies. In particular,quotients of completely regular spaces need not beregular. Quotients of Tychonoff spaces need not even beHausdorff, with one elementary counterexample being theline with two origins. There are closed quotients of theMoore plane that provide counterexamples.

For any topological space${\displaystyle X,}$ let${\displaystyle C(X)}$ denote the family of real-valuedcontinuous functions on${\displaystyle X}$ and let${\displaystyle C_b(X)}$ be the subset ofbounded 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 theinitial 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 ofzero 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 thecozero sets of${\displaystyle X}$ form abasis 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${\displaystyle C_{\tau }(X)}$ or, equivalently, the topology generated by the basis of cozero sets in${\displaystyle (X,\tau ).}$ Then ρ will be thefinest completely regular topology on${\displaystyle X}$ that is coarser than${\displaystyle \tau .}$ This construction isuniversal in the sense that any continuous function$${\displaystyle f:(X,\tau )\to Y}$$to a completely regular space${\displaystyle Y}$ will be continuous on${\displaystyle (X,\rho ).}$ In the language ofcategory theory, thefunctor that sends${\displaystyle (X,\tau )}$ to${\displaystyle (X,\rho )}$ isleft adjoint to the inclusion functorCReg →Top. Thus the category of completely regular spacesCReg is areflective subcategory ofTop, thecategory of topological spaces. By takingKolmogorov quotients, one sees that the subcategory of Tychonoff spaces is also reflective.

One can show that${\displaystyle C_{\tau }(X)=C_{\rho }(X)}$ in the above construction so that the rings${\displaystyle C(X)}$ and${\displaystyle C_b(X)}$ are typically only studied for completely regular spaces${\displaystyle X.}$

The category ofrealcompact Tychonoff spaces is anti-equivalent to the category of the rings${\displaystyle C(X)}$ (where${\displaystyle X}$ is realcompact) together with ring homomorphisms as maps. For example one can reconstruct${\displaystyle X}$ from${\displaystyle C(X)}$ when${\displaystyle X}$ is (real) compact. 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 inreal algebraic geometry, is the class ofreal closed rings.

Tychonoff spaces are precisely those spaces that can beembedded incompact Hausdorff spaces. More precisely, for every Tychonoff space${\displaystyle X,}$ there exists a compact Hausdorff space${\displaystyle K}$ such that${\displaystyle X}$ ishomeomorphic to a subspace of${\displaystyle K.}$

In fact, one can always choose${\displaystyle K}$ to be aTychonoff cube (i.e. a possibly infinite product ofunit intervals). Every Tychonoff cube is compact Hausdorff as a consequence ofTychonoff's theorem. Since every subspace of a compact Hausdorff space is Tychonoff one has:

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

Of particular interest are those embeddings where the image of${\displaystyle X}$ isdense in${\displaystyle K;}$ these are called Hausdorffcompactifications of${\displaystyle X.}$Given any embedding of a Tychonoff space${\displaystyle X}$ in a compact Hausdorff space${\displaystyle K}$ theclosure of the image of${\displaystyle X}$ in${\displaystyle K}$ is a compactification of${\displaystyle X.}$In the same 1930 article^[2] where Tychonoff defined completely regular spaces, he also proved that every Tychonoff space has a Hausdorff compactification.

Among those Hausdorff compactifications, there is a unique "most general" one, theStone–Čech compactification${\displaystyle \beta X.}$It is characterized by theuniversal property that, given a continuous map${\displaystyle f}$ from${\displaystyle X}$ to any other compact Hausdorff space${\displaystyle Y,}$ there is aunique continuous map${\displaystyle g:\beta X\to Y}$ that extends${\displaystyle f}$ in the sense that${\displaystyle f}$ is thecomposition of${\displaystyle g}$ and${\displaystyle j.}$

Complete regularity is exactly the condition necessary for the existence ofuniform structures on a topological space. In other words, everyuniform space has a completely regular topology and every completely regular space${\displaystyle X}$ isuniformizable. A topological space admits a separated uniform structure if and only if it is Tychonoff.

Given a completely regular space${\displaystyle X}$ there is usually more than one uniformity on${\displaystyle X}$ that is compatible with the topology of${\displaystyle X.}$ However, there will always be a finest compatible uniformity, called thefine uniformity on${\displaystyle X.}$ If${\displaystyle X}$ is Tychonoff, then the uniform structure can be chosen so that${\displaystyle \beta X}$ becomes thecompletion of the uniform space${\displaystyle X.}$

• Stone–Čech compactification – Concept in topology

• Separation axioms
• Topological spaces
• Topology

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