| Topological spaces in separation axiom |
| Kolmogorov (T0) version |
|---|
| T0 | T1 | T2 | T2½ | completely T2 T3 | T3½ | T4 | T5 | T6 |
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 mathematical structures that allow the formal definition of 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 In Topology and related branches of Mathematics, T1 spaces and R0 spaces are particular kinds of Topological spaces The In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Topology, completely Hausdorff spaces and Urysohn (or T2½) spaces are types of Topological spaces satisfying slightly In Topology and related fields of Mathematics, regular spaces and T3 spaces are particularly convenient 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, T5 spaces, and T6 spaces In Topology and related branches of Mathematics, normal spaces, T4 spaces, T5 spaces, and T6 spaces In Topology and related branches of Mathematics, normal spaces, T4 spaces, T5 spaces, and T6 spaces Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Topological spaces are mathematical structures that allow the formal definition of 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, separated space or T2 space is a Topological space
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Definitions
Suppose that X is a topological space. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Let x and y be points in X.
- 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 = ∅). 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 Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. (Note that a "closed neighborhood of x" is a closed set that contains an open set containing x. In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in )
- 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. 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 Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, the unit interval is the interval, that is the set of all Real numbers x such that zero is less than or equal to x
A Urysohn space, or T2½ space, is a space in which any two distinct points can be separated by closed neighborhoods.
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, separated space or T2 space is a Topological space
One can also show that every regular Hausdorff space is Urysohn and every Tychonoff space (=completely regular Hausdorff space) is completely Hausdorff. In Topology and related fields of Mathematics, regular spaces and T3 spaces are particularly convenient 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 summary we have the following implications:
| Tychonoff (T3½) |  |
regular Hausdorff (T3) | ||||
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| 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 convenient kinds of Topological spaces In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Topology and related branches of Mathematics, T1 spaces and R0 spaces are particular kinds of Topological spaces The
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. In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets The cocountable topology or countable complement topology on any set X consists of the Empty set and all Cocountable subsets of X, 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. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in This space is completely Hausdorff and Urysohn, but not regular (and thus not Tychonoff).
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
- Lynn Arthur Steen and J. PlanetMath is a free, collaborative online Mathematics Encyclopedia. Arthur Seebach, Jr. , Counterexamples in Topology. Counterexamples in Topology (1970 2nd ed 1978 is a book on Mathematics by topologists Lynn Steen and J 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).
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