Hausdorff space (plural Hausdorff spaces)
(topology) A topological space in which for any two distinct points x and y, there is a pair of disjoint open sets U and V such that
x
∈
U
{\displaystyle x\in U}
and
y
∈
V
{\displaystyle y\in V}
.
Also, a subset of a normal space need not be normal (i.e. not every normal Hausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čech compactification (which is normal Hausdorff). Source: Internet
Although Hausdorff spaces are not, in general, regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular. Source: Internet
Although Hausdorff spaces aren't generally regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular. Source: Internet
Any compact Hausdorff space is a complete uniform space with respect to the unique uniformity compatible with the topology. Source: Internet
But in fact, there is a simpler method available in the locally compact case; the one-point compactification will embed X in a compact Hausdorff space a(X) with just one extra point. Source: Internet
Commutative C*-algebras Let X be a locally compact Hausdorff space. Source: Internet