Hausdorff space

A Hausdorff space (also Hausdorff space shear ) (after Felix Hausdorff ) or separated space is a topological space in which the separation axiom (also Hausdorf fig stem or Hausdorff'sches separation axiom called ) applies.

Definition

A topological space has the Hausdorf fig stem, if for all disjoint with open environments and exist.

In other words, all pairwise distinct points of are separated by environments. A topological space that satisfies the Hausdorf fig stem, Hausdorff space is called.

Position in the hierarchy of topological spaces

A topological space is then a Hausdorff space if it is präregulär ():

And the Kolmogorov - property () has:

Topologically distinguishable names of two points if and only if there is an open set containing the point a, the other is not.

Proof:

• If and are given, followed immediately: this conclusion can be drawn purely formal, without knowing what's called topologically distinguishable at all.
• The opposite conclusion from on and goes like this: From the definition of follows for different, the existence of the amount, but does not contain, ergo applies.
• Be, two topologically distinguishable points: then there is not a lot that contains the one point, but the other; is thus. Then follows with, and that are separated by environments. Ergo applies.

Specialization

A Hausdorff space which is additionally normal is called a T4 - space.

Examples

Specifically, in a topological Hausdorff space limits - unlike in general topological spaces - unique.

Just about all considered in the analysis spaces are Hausdorff spaces. In particular, each metric space is a Hausdorff space.

An example of a Hausdorff space which is not a metric space is the set of countable ordinals with the usual order topology.

If the spectrum of a ring with the Zariski topology provided, you get a sober topological space, which is usually not präregulär, let 's the Hausdorff.

Many examples of non- Hausdorff spaces is obtained as a quotient space of manifolds with respect to some group effects or more general equivalence relations. For example, the leaf area of ​​Reeb - foliation (ie the quotient space with respect to the equivalence relation. : Two points are equivalent iff they belong to the same leaf ) is not Hausdorff.

Lokaleuklidische spaces need not be Hausdorff. The resultant of two copies of identification through an open interval space is locally homeomorphic to but not Hausdorff.