Editing Tychonoff space (section)

==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>