How Do Completely Regular Hausdorff Spaces Relate to Compact Hausdorff Spaces?

[aliceinwonder]

Is there any inclusion relationship between completely regular Hausdorff space and compact Haudorff space?
What is the example to show their inclusion relationship?

Thanks.

 

[morphism]

A compact Hausdorff space is normal so, by Urysohn's lemma, completely regular.

Completely regular Hausdorff spaces need not be compact: just take any non-compact metric space!

 

[mathwonk]

a completely regular hausdorff space is a subspace of a compact hausdorff space. for each completely regular such space, there is a one one correspondence between compact hausdorff spaces containing it and uniformly closed subalgebras of its algebra of continuous functions which are point separating and contain the constants.

 

[mathwonk]

for example the plane is a completely regular T2 space, and if we consider the algebra of functions which have limits at infinity, the corresponding compact space containing it is homeomorphic to the sphere, i.e. to the plane plus one point at infinity where those functiions attain their limit as a value.

I.e. the correspondence is that a subalgebra of functions gives rise to a compact space on which each of those functions extends to a continuous function. it follows in pARTICULAr that there is a huge compact space to which every function continuous on the original space extends. this is called the stone cech compactification.