*( BCT2 ) Every locally compact Hausdorff space is a Baire space.

*( BCT2 ) Every locally compact Hausdorff space is a Baire space.

Therefore, the previous property tells us that if G is a σ-compact, Hausdorff topological group that is also a Baire space, then G is locally compact.

This shows that for Hausdorff topological groups that are also Baire spaces, σ-compactness implies local compactness.

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of X ; this is known as the σ-algebra of Borel sets.

The real line R with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it.

The prototypical example of a Banach algebra is, the space of ( complex-valued ) continuous functions on a locally compact ( Hausdorff ) space that vanish at infinity.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

* Uniform algebra: A Banach algebra that is a subalgebra of C ( X ) with the supremum norm and that contains the constants and separates the points of X ( which must be a compact Hausdorff space ).

For ƒ ∈ C ( X ) ( with a compact Hausdorff space X ), one sees that:

Equipped with the topology of pointwise convergence on A ( i. e., the topology induced by the weak -* topology of A < sup >∗</ sup >), the character space, Δ ( A ), is a Hausdorff compact space.

* Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification.

Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.

* The structure space of a commutative unital Banach algebra is a compact Hausdorff space.

By the same construction, every locally compact Hausdorff space X is a dense subspace of a compact Hausdorff space having at most one point more than X ..

* A compact subset of a Hausdorff space is closed.

This is to say, compact Hausdorff space is normal.

( Gelfand – Naimark theorem ) Properties of the Banach space of continuous functions on a compact Hausdorff space are central to abstract analysis.

* Every continuous map from a compact space to a Hausdorff space is closed and proper ( i. e., the pre-image of a compact set is compact.

) In particular, every continuous bijective map from a compact space to a Hausdorff space is a homeomorphism.

** Hausdorff maximal principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.

F is a constant and D is a parameter that Richardson found depended on the coastline approximated by L. He gave no theoretical explanation but Mandelbrot identified L with a non-integer form of the Hausdorff dimension, later the fractal dimension.

* The spectrum of any commutative ring with the Zariski topology ( that is, the set of all prime ideals ) is compact, but never Hausdorff ( except in trivial cases ).

* A topological space can be embedded in a compact Hausdorff space if and only if it is a Tychonoff space.

A Hausdorff space is H-closed if every open cover has a finite subfamily whose union is dense.

Since every compact Hausdorff space is a Tychonoff space, and every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space.

In fact, the converse is also true ; being a Tychonoff space is both necessary and sufficient for possessing a Hausdorff compactification.

The following identity (" Hadamard Lemma ") involving nested commutators, underlying the Campbell – Baker – Hausdorff expansion of log ( expA expB ), is also useful:

This condition is the third separation axiom ( after T < sub > 0 </ sub > and T < sub > 1 </ sub >), which is why Hausdorff spaces are also called T < sub > 2 </ sub > spaces.

The principle is also called the Hausdorff maximality theorem or the Kuratowski lemma ( Kelley 1955: 33 ).

In mathematics, the Hausdorff dimension ( also known as the Hausdorff – Besicovitch dimension ) is an extended non-negative real number associated with any metric space.

Formally, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space.

Although not part of this definition, many authors require that the topology on G be Hausdorff ; this corresponds to the identity map being a closed inclusion ( hence also a cofibration ).

A normal Hausdorff space is also called a T < sub > 4 </ sub > space.

In almost all applications, locally compact spaces are also Hausdorff, and this article is thus primarily concerned with locally compact Hausdorff ( LCH ) spaces.

Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article compact space.

As mentioned in the following section, no Hausdorff space can possibly be locally compact if it is not also a Tychonoff space ; there are some examples of Hausdorff spaces that are not Tychonoff spaces in that article.

In the case where is locally compact, e. g. or, the image of forms an open subset of, or indeed of any compactification, ( this is also a necessary condition, as an open subset of a compact Hausdorff space is locally compact ).

Since a Hausdorff space is the same as a preregular T < sub > 0 </ sub > space, a regular space that is also T < sub > 0 </ sub > must be Hausdorff ( and thus T < sub > 3 </ sub >).

On the other hand, those results that are truly about regularity generally don't also apply to nonregular Hausdorff spaces.

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.

Note that neither of these statements implies the other, since there are complete metric spaces which are not locally compact ( the irrational numbers with the metric defined below ; also, any Banach space of infinite dimension ), and there are locally compact Hausdorff space which are not metrizable ( for instance, any uncountable product of non-trivial compact Hausdorff spaces is such ; also, several function spaces used in Functional Analysis ; the uncountable Fort space ).