Editing Tychonoff space

{{Short description|Type of regular Hausdorff space}}{{Separation axioms}}

In [[topology]] and related branches of [[mathematics]], '''Tychonoff spaces''' and '''completely regular spaces''' are kinds of [[topological space]]s.  These conditions are examples of [[separation axiom]]s. 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<ref>{{cite journal |last1=Urysohn |first1=Paul |title=Über die Mächtigkeit der zusammenhängenden Mengen |journal=Mathematische Annalen |date=1925 |volume=94 |issue=1 |pages=262–295 |doi=10.1007/BF01208659}} See pages 291 and 292.</ref> without giving it a name.  But it was [[Andrey Tychonoff]] who introduced the terminology ''completely regular'' in 1930.<ref name="tychonoff-1930">{{cite journal |last1=Tychonoff |first1=A. |title=Über die topologische Erweiterung von Räumen |journal=Mathematische Annalen |date=1930 |volume=102 |issue=1 |pages=544–561 |doi=10.1007/BF01782364}}</ref>

==Definitions==
[[File:Separation of a point from a closed set via a continuous function.svg|thumb|300x300px|Separation of a point from a closed set via a continuous function.]]
A topological space <math>X</math> is called '''{{em|completely regular}}''' if points can be [[Separated sets|separated]] from closed sets via (bounded) continuous real-valued functions. In technical terms this means: for any [[closed set]] <math>A \subseteq X</math> and any [[Point (geometry)|point]] <math>x \in X \setminus A,</math> there exists a [[real line|real-valued]] [[continuous function (topology)|continuous function]] <math>f : X \to \R</math> such that <math>f(x)=1</math> and <math>f\vert_{A} = 0.</math> (Equivalently one can choose any two values instead of <math>0</math> and <math>1</math> and even require that <math>f</math> be a bounded function.)

A topological space is called a '''{{em|Tychonoff space}}''' (alternatively: '''{{em|T<sub>3½</sub> space}}''', or {{em|T<sub>π</sub> space}}, or {{em|completely T<sub>3</sub> 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 [[Kolmogorov space|T<sub>0</sub>]]. 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 space|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 space|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 space|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'',{{sfn|Willard|1970|loc=Problem 18G}}{{sfn|Steen|Seebach|1995|loc=Example 90}} 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'',{{sfn|Steen|Seebach|1995|loc=Example 92}}<ref>{{cite journal |last1=Hewitt |first1=Edwin |title=On Two Problems of Urysohn |journal=Annals of Mathematics |date=1946 |volume=47 |issue=3 |pages=503–509 |doi=10.2307/1969089|jstor=1969089 }}</ref> 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 topology|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 (topology)|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 topology|final topologies]]. In particular, [[Quotient space (topology)|quotients]] of completely regular spaces need not be [[Regular space|regular]]. Quotients of Tychonoff spaces need not even be [[Hausdorff space|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 <math>X,</math> let <math>C(X)</math> denote the family of real-valued [[Continuous function (topology)|continuous functions]] on <math>X</math> and let <math>C_b(X)</math> be the subset of [[Bounded function|bounded]] real-valued continuous functions.

Completely regular spaces can be characterized by the fact that their topology is completely determined by <math>C(X)</math> or <math>C_b(X).</math> In particular:

* A space <math>X</math> is completely regular if and only if it has the [[initial topology]] induced by <math>C(X)</math> or <math>C_b(X).</math>
* A space <math>X</math> is completely regular if and only if every closed set can be written as the intersection of a family of [[zero set]]s in <math>X</math> (i.e. the zero sets form a basis for the closed sets of <math>X</math>).
* A space <math>X</math> is completely regular if and only if the [[cozero set]]s of <math>X</math> form a [[Basis (topology)|basis]] for the topology of <math>X.</math>

Given an arbitrary topological space <math>(X, \tau)</math> there is a universal way of associating a completely regular space with <math>(X, \tau).</math> Let ρ be the initial topology on <math>X</math> induced by <math>C_{\tau}(X)</math> or, equivalently, the topology generated by the basis of cozero sets in <math>(X, \tau).</math> Then ρ will be the [[Finest topology|finest]] completely regular topology on <math>X</math> that is coarser than <math>\tau.</math> This construction is [[Universal property|universal]] in the sense that any continuous function
<math display=block>f : (X, \tau) \to Y</math>
to a completely regular space <math>Y</math> will be continuous on <math>(X, \rho).</math> In the language of [[category theory]], the [[functor]] that sends <math>(X, \tau)</math> to <math>(X, \rho)</math> is [[left adjoint]] to the inclusion functor '''CReg''' &rarr; '''Top'''. Thus the category of completely regular spaces '''CReg''' is a [[reflective subcategory]] of '''Top''', the [[category of topological spaces]]. By taking [[Kolmogorov quotient]]s, one sees that the subcategory of Tychonoff spaces is also reflective.

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

The category of [[Realcompact space|realcompact]] Tychonoff spaces is anti-equivalent to the category of the rings <math>C(X)</math> (where <math>X</math> is realcompact) together with ring homomorphisms as maps. For example one can reconstruct <math>X</math> from <math>C(X)</math> when <math>X</math> 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 in [[real algebraic geometry]], is the class of [[real closed ring]]s.

===Embeddings===

Tychonoff spaces are precisely those spaces that can be [[Topological embedding|embedded]] in [[compact Hausdorff space]]s. More precisely, for every Tychonoff space <math>X,</math> there exists a compact Hausdorff space <math>K</math> such that <math>X</math> is [[homeomorphic]] to a subspace of <math>K.</math>

In fact, one can always choose <math>K</math> to be a [[Tychonoff cube]] (i.e. a possibly infinite product of [[unit interval]]s). Every Tychonoff cube is compact Hausdorff as a consequence of [[Tychonoff'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''.

===Compactifications===

Of particular interest are those embeddings where the image of <math>X</math> is [[Dense subset|dense]] in <math>K;</math> these are called Hausdorff [[Compactification (mathematics)|compactifications]] of <math>X.</math>
Given any embedding of a Tychonoff space <math>X</math> in a compact Hausdorff space <math>K</math> the [[Closure (topology)|closure]] of the image of <math>X</math> in <math>K</math> is a compactification of <math>X.</math>
In the same 1930 article<ref name="tychonoff-1930"/> 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, the [[Stone–Čech compactification]] <math>\beta X.</math>
It is characterized by the [[universal property]] that, given a continuous map <math>f</math> from <math>X</math> to any other compact Hausdorff space <math>Y,</math> there is a [[Unique (mathematics)|unique]] continuous map <math>g : \beta X \to Y</math> that extends <math>f</math> in the sense that <math>f</math> is the [[Composition (functions)|composition]] of <math>g</math> and <math>j.</math>

===Uniform structures===

Complete regularity is exactly the condition necessary for the existence of [[uniform structure]]s on a topological space. In other words, every [[uniform space]] has a completely regular topology and every completely regular space <math>X</math> is [[uniformizable]]. A topological space admits a separated uniform structure if and only if it is Tychonoff.

Given a completely regular space <math>X</math> there is usually more than one uniformity on <math>X</math> that is compatible with the topology of <math>X.</math> However, there will always be a finest compatible uniformity, called the [[fine uniformity]] on <math>X.</math> If <math>X</math> is Tychonoff, then the uniform structure can be chosen so that <math>\beta X</math> becomes the [[Completion (topology)|completion]] of the uniform space <math>X.</math>

==See also==

* {{annotated link|Stone–Čech compactification}}

==Citations==
{{reflist}}

==Bibliography==

{{refbegin}}
* {{Cite book| title = Rings of continuous functions | edition = Dover reprint
 | last1 = Gillman | first1 = Leonard
 | last2 = Jerison | first2 = Meyer
 | author1-link = Leonard Gillman
 | year = 1960
 | publisher = Springer-Verlag | location = NY
 | series = Graduate Texts in Mathematics, No. 43
 | url = https://books.google.com/books?id=QPQ_DwAAQBAJ
 | page = xiii
 | isbn = 978-048681688-3
}}
*{{Cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | orig-year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446  | year=1995 }}
* {{Cite book| title = General Topology | edition = Dover reprint
 | last = Willard | first = Stephen | year = 1970
 | publisher = Addison-Wesley Publishing Company | location = Reading, Massachusetts
 | url = https://books.google.com/books?id=UrsHbOjiR8QC
 | isbn = 0-486-43479-6
}}
{{refend}}

[[Category:Separation axioms]]
[[Category:Topological spaces]]
[[Category:Topology]]