Urysohn space: Difference between revisions

Latest revision as of 23:00, 27 January 2012

Definition

A topological space $X$ is termed a Urysohn space if, for any two distinct points $x, y \in X$ , there exist disjoint open subsets $U ∋ x, V ∋ y$ such that the closures <math\overline{U}</math> and $\bar{V}$ are disjoint closed subsets of $X$ .

Note that the term Urysohn space is also used for the somewhat stronger notion of functionally Hausdorff space. There is a terminological ambiguity here.

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

In the T family (properties of topological spaces related to separation axioms), this is called: T2.5

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
regular Hausdorff space (also called $T_{3}$ )	T1 and any point can be separated from any disjoint closed subset	regular Hausdorff implies Urysohn	Urysohn not implies regular	\|FULL LIST, MORE INFO
functionally Hausdorff space				\|FULL LIST, MORE INFO
Tychonoff space (also called $T_{3.5}$ )	T1 and any point and disjoint closed subset can be separated by a continuous function			Functionally Hausdorff space\|FULL LIST, MORE INFO
normal Hausdorff space (also called $T_{4}$ )	T1 and normal			Functionally Hausdorff space\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Hausdorff space				\|FULL LIST, MORE INFO
T1 space				Hausdorff space\|FULL LIST, MORE INFO
Kolmogorov space				Hausdorff space\|FULL LIST, MORE INFO

@@ Line 23: / Line 23: @@
 |-
 | [[Weaker than::normal Hausdorff space]] (also called <math>T_4</math>) || T1 and [[normal space|normal]] || || || {{intermediate notions short|Urysohn space|normal Hausdorff space}}
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::Hausdorff space]] || || || || {{intermediate notions short|Hausdorff space|Urysohn space}}
+|-
+| [[Stronger than::T1 space]] || || || || {{intermediate notions short|T1 space|Urysohn space}}
+|-
+| [[Stronger than::Kolmogorov space]] || || || || {{intermediate notions short|Kolmogorov space|Urysohn space}}
 |}