Hausdorff in a sentence

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).

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.

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.

A topological group G is Hausdorff if and only if the trivial one-element subgroup is closed in G. If G is not Hausdorff then one can obtain a Hausdorff group by passing to the quotient space G/K where K is the closure of the identity.

However, the compact Hausdorff spaces are " absolutely closed ", in the sense that, if you embed a compact Hausdorff space K in an arbitrary Hausdorff space X, then K will always be a closed subset of X; the "surrounding space" does not matter here.

Siehe Findbuch Nachlass Hausdorff Work and reception Hausdorff as philosopher and writer (Paul Mongré) His volume of aphorisms in 1897 was the first published work of Hausdorff under the pseudonym Paul Mongré.

Given the historical confusion of the meaning of the terms, verbal descriptions when applicable are helpful, that is, "normal Hausdorff" instead of "T 4 ", or "completely normal Hausdorff" instead of "T 5 ".

Hausdorff dimension and Minkowski dimension The Minkowski dimension is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations.

Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom.

He was selflessly supported at this time by Erich Bessel-Hagen, a loyal friend to the Hausdorff family who obtained books and magazines from the Library of the Institute, which Hausdorff was no longer allowed to enter as a Jew.

In 1904 Hausdorff published the recursion named after him: For each non-limit ordinal we have This formula was, together with the later notion of cofinality introduced by Hausdorff, the basis for all further results for Aleph exponentiation.

Indeed, when analysts run across a non-Hausdorff space, it is still probably at least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff.

Proof that paracompact Hausdorff spaces admit partitions of unity A Hausdorff space is paracompact if and only if it every open cover admits a subordinate partition of unity.

The last fact follows from f(X) being compact Hausdorff, and hence (since compact metrisable spaces are necessarily second countable); as well as the fact that compact Hausdorff spaces are metrisable exactly in case they are second countable.

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

Vickers (1989) p.65 H ; Hausdorff : A Hausdorff space (or T 2 space) is one in which every two distinct points have disjoint neighbourhoods.

About the humiliations to which Hausdorff and his family especially were exposed to after Kristallnacht in 1938, much is known and from many different sources, such as from the letters of Bessel-Hagen.

A characterization of finite dimensionality is that a Hausdorff TVS is locally compact if and only if it is finite-dimensional (therefore isomorphic to some Euclidean space).

After his habilitation, Hausdorff wrote another work on optics, on non-Euclidean geometry, and on hypercomplex number systems, as well as two papers on probability theory.

A Hausdorff uniform space is metrizable if its uniformity can be defined by a countable family of pseudometrics.