Editing Tychonoff space (section)

==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.