In topology, completely Hausdorff spaces and Urysohn spaces are types of topological spaces satisfying slightly stronger separation axioms than the more familiar Hausdorff space. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. ... In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces, named after Andrey Kolmogorov, form a broad class of well-behaved topological spaces. ... The title given to this article is incorrect due to technical limitations. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... The definition used is this article is in contradiction with the usage of the term elsewhere in Wikipedia. ... In topology and related fields of mathematics, regular spaces and T3 spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

Definitions

Suppose that X is a topological space. Let x and y be points in X. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

• We say that x and y can be separated by closed neighborhoods if there exists a closed neighborhood U of x and a closed neighborhood V of y such that U and V are disjoint (U ∩ V = ∅).
• We say that x and y can be separated by a function if there exists a continuous function f : X → [0,1] (the unit interval) with f(x) = 0 and f(y) = 1.

A Urysohn space, or T_2½ space, is a space in which any two distinct points can be separated by closed neighborhoods. In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, two sets are said to be disjoint if they have no element in common. ... In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way. ... In topology, a continuous function is generally defined as one for which preimages of open sets are open. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...

A completely Hausdorff space, or functionally Hausdorff space, is a space in which any two distinct points can be separated by a function.

Naming conventions

The study of separation axioms is notorious for conflicts with naming conventions used. The definitions used in this article are those given by Willard (1970) and are the more modern definitions. Steen and Seebach (1970) and various other authors reverse the definition of completely Hausdorff spaces and Urysohn spaces. Readers of textbooks in topology must be sure to check the definitions used by the author. See History of the separation axioms for more on this issue. In general topology, the separation axioms have had a convoluted history, with many competing meanings for the same term, and many competing terms for the same concept. ...

Relation to other separation axioms

It is an easy exercise to show that any two points which can be separated by a function can be separated by closed neighborhoods. If they can be separated by closed neighborhoods then clearly they can be separated by neighborhoods. It follows that every completely Hausdorff space is Urysohn and every Urysohn space is Hausdorff. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

One can also show that every regular Hausdorff space is Urysohn and every Tychonoff space (=completely regular Hausdorff space) is completely Hausdorff. In summary we have the following implications: In topology and related fields of mathematics, regular spaces and T3 spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ...

Tychonoff (T3½)		regular Hausdorff (T3)
completely Hausdorff		Urysohn (T2½)		Hausdorff (T2)		T1

One can find counterexamples showing that none of these implications reverse^[1]. In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ... In topology and related fields of mathematics, regular spaces and T3 spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... The title given to this article is incorrect due to technical limitations. ...

Examples

The cocountable extension topology is the topology on the real line generated by the union of the usual Euclidean topology and the cocountable topology. Sets are open in this topology if and only if they are of the form U A where U is open in the Euclidean topology and A is countable. This space is completely Hausdorff and Urysohn, but not regular (and thus not Tychonoff). In mathematics, the real line is simply the set of real numbers. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a cocountable subset of a set X is a subset Y whose complement in X is a countable set. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics the term countable set is used to describe the size of a set, e. ...

There are obscure examples of spaces which are Hausdorff but not Urysohn, and spaces which are Urysohn but not completely Hausdorff or regular Hausdorff. For details see Steen and Seebach.

References

1. ^ Hausdorff space not completely Hausdorff on PlanetMath

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).
• Stephen Willard, General Topology, Addison-Wesley, 1970. Reprinted by Dover Publications, New York, 2004. ISBN 0-486-43479-6 (Dover edition).

• Completely Hausdorff on PlanetMath