Complete Hausdorff room

Complete Hausdorff spaces are in the topology and related areas of mathematics such topological spaces is based on their values under whose points real-valued continuous functions can be distinguished.

definition

Be a topological space. We say that two points and are separated by a function if there is a continuous function such that and . ${\ displaystyle X}$ ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle f \ colon X \ rightarrow [0,1]}$ ${\ displaystyle f (x) = 0}$ ${\ displaystyle f (y) = 1}$

${\ displaystyle X}$ is a complete Hausdorff space if two different points and are always separated by a function. It is also said to be complete . In other words, the set of all continuous -valent functions points separating . ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle X}$ ${\ displaystyle T_ {2}}$ ${\ displaystyle [0,1]}$

Relationships with the other axioms of separation

Every complete Hausdorff space is a Urysohn space and thus fulfills the axioms of separation , and, among other things . ${\ displaystyle T_ {0}}$ ${\ displaystyle T_ {1}}$ ${\ displaystyle T_ {2}}$

On the other hand, every Tychonoff room is a complete Hausdorff room.

On the other hand, there are examples that show that neither every complete Hausdorff space is a regular Hausdorff space , nor that every regular Hausdorff space is a complete Hausdorff space.

Examples

The Euclidean topology defines a complete Hausdorff space. ${\ displaystyle \ mathbb {R} ^ {n}}$

We define the topology that is generated by the union of the amount topology with the topology whose open sets are the sets of the form with an open set in the amount topology and a countable set . As an extension of the amount topology, this topology is completely Hausdorffian. But it is not regular and therefore we do not get a Tychonoff space. ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle U \ setminus A}$ ${\ displaystyle U}$ ${\ displaystyle A}$

Relationship with Stone-Čech compactification

The canonical mapping of a topological space in its Stone-Čech compactification is injective if and only if it is completely Hausdorffian. ${\ displaystyle X}$ ${\ displaystyle X}$

Individual evidence

^ Nicolas Bourbaki : Eléments de mathématique. General topology. Ch. 1 à 4. Reimpression inchangée de l'edition originale de 1971. Springer, Berlin et al. 2007, ISBN 978-3-540-33936-6 , chapter 9, p. 10.

[1] Nicolas Bourbaki : Eléments de mathématique. General topology. Ch. 1 à 4. Reimpression inchangée de l'edition originale de 1971. Springer, Berlin et al. 2007, ISBN 978-3-540-33936-6 , chapter 9, p. 10.