Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact.

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

Stone spaces, compact totally disconnected Hausdorff spaces, form the abstract framework in which these spectra are studied.

; Hausdorff spaces

* Two compact Hausdorff spaces X < sub > 1 </ sub > and X < sub > 2 </ sub > are homeomorphic if and only if their rings of continuous real-valued functions C ( X < sub > 1 </ sub >) and C ( X < sub > 2 </ sub >) are isomorphic.

Embeddings into compact Hausdorff spaces may be of particular interest.

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.

For example, they are Hausdorff paracompact spaces ( and hence normal and Tychonoff ) and first-countable.

Quotients of Tychonoff spaces need not even be Hausdorff.

embedded in compact Hausdorff spaces.

The theorem generalizes Urysohn's lemma and is widely applicable, since all metric spaces and all compact Hausdorff spaces are normal.

The two definitions are equivalent for many well-behaved spaces, including all Hausdorff σ-compact spaces, but can be different in more pathological spaces.

* a characterization of Hausdorff spaces which are now called Kuratowski closure axioms ;

These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T < sub > 5 </ sub > spaces, and perfectly normal Hausdorff spaces, or T < sub > 6 </ sub > spaces.

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

A topological space is Hausdorff if and only if it is both preregular ( i. e. topologically distinguishable points are separated by neighbourhoods ) and Kolmogorov ( i. e. distinct points are topologically distinguishable ).

The core motivating idea are the various Fourier transforms, which can be generalized to a transform of functions defined on Hausdorff locally compact topological groups.

There are, however, many irregular sets that have noninteger Hausdorff dimension.

For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional geometry and science, the Hausdorff dimension is an integer.

But Benoît Mandelbrot observed that fractals, sets with noninteger Hausdorff dimensions, are found everywhere in nature.

These notions ( packing dimension, Hausdorff dimension, Minkowski – Bouligand dimension ) all give the same value for many shapes, but there are well documented exceptions.

* For a compact Hausdorff space X, the following are equivalent:

Of particular interest are those embeddings where the image of X is dense in K ; these are called Hausdorff compactifications of X.

Spec ( R ) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in R are precisely the closed points in this topology.

It is named for Henry Frederick Baker, John Edward Campbell, and Felix Hausdorff.

* Felix Hausdorff ( 1868 – 1942 ), the German mathematician after whom Hausdorff spaces are named

It is named after Felix Hausdorff.

In mathematics, Gromov – Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.

In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in to each set in R < sup > n </ sup > or, more generally, in any metric space.

In mathematics ( specifically, measure theory ), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.

In mathematics, the Hausdorff paradox, named after Felix Hausdorff, states that if you remove a certain countable subset of the sphere S < sup > 2 </ sup >, the remainder can be divided into three disjoint subsets A, B and C such that A, B, C and B ∪ C are all congruent.

In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the theory of continuous functions on H. The concept is named after Harald Bohr who pioneered the study of almost periodic functions, on the real line.

* 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 ).

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

Very shortly after that work was submitted, by March 1918, Felix Hausdorff expanded the definition of " dimension ", significantly for the evolution of the definition of fractals, to allow for sets to have noninteger dimensions.

A straight line, for instance, is self-similar but not fractal because it lacks detail, is easily described in Euclidean language, has the same Hausdorff dimension as topological dimension, and is fully defined without a need for recursion.

That is, the Hausdorff dimension of an n-dimensional inner product space equals n. This means, for example, the Hausdorff dimension of a point is zero, the Hausdorff dimension of a line is one, and the Hausdorff dimension of the plane is two.

Many of the technical developments used to compute the Hausdorff dimension for highly irregular sets were obtained by Abram Samoilovitch Besicovitch.

The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the Baker – Campbell – Hausdorff formula: there exists a neighborhood U of the zero element of, such that for u, v in U we have

More precisely, for every Tychonoff space X, there exists a compact Hausdorff space K such that X is homeomorphic to a subspace of K.

In the mathematical discipline of general topology, Stone – Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX.

As is usual for universal properties, this universal property, together with the fact that βX is a compact Hausdorff space containing X, characterizes βX up to homeomorphism.

Some authors add the assumption that the starting space be Tychonoff ( or even locally compact Hausdorff ), for the following reasons:

where the product is over all maps from X to compact Hausdorff spaces C. This works intuitively but fails for the technical reason that the collection of all such maps is a proper class rather than a set.

There are several ways to modify this idea to make it work ; for example, one can restrict the compact Hausdorff spaces C to have underlying set P ( P ( X )) ( the power set of the power set of X ), which is sufficiently large that it has cardinality at least equal to that of every compact Hausdorff set to which X can be mapped with dense image.