Editing Tychonoff space (section)

===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>