Urysohn space: Difference between revisions
No edit summary |
|||
| (2 intermediate revisions by the same user not shown) | |||
| Line 3: | Line 3: | ||
A [[topological space]] <math>X</math> is termed a '''Urysohn space''' if, for any two ''distinct'' points <math>x,y \in X</matH>, there exist disjoint [[open subset]]s <math>U \ni x, V \ni y </math> such that the [[closure]]s <math\overline{U}</math> and <math>\overline{V}</math> are disjoint [[closed subset]]s of <math>X</math>. | A [[topological space]] <math>X</math> is termed a '''Urysohn space''' if, for any two ''distinct'' points <math>x,y \in X</matH>, there exist disjoint [[open subset]]s <math>U \ni x, V \ni y </math> such that the [[closure]]s <math\overline{U}</math> and <math>\overline{V}</math> are disjoint [[closed subset]]s of <math>X</math>. | ||
Note that the term Urysohn space is also used for the somewhat stronger notion of [[ | Note that the term Urysohn space is also used for the somewhat stronger notion of [[functionally Hausdorff space]]. There is a terminological ambiguity here. | ||
{{topospace property}} | {{topospace property}} | ||
| Line 10: | Line 10: | ||
==Relation with other properties== | ==Relation with other properties== | ||
===Stronger properties=== | |||
{| class="sortable" border="1" | {| class="sortable" border="1" | ||
| Line 21: | 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 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}} | |||
|} | |} | ||
Latest revision as of 23:00, 27 January 2012
Definition
A topological space is termed a Urysohn space if, for any two distinct points , there exist disjoint open subsets such that the closures <math\overline{U}</math> and are disjoint closed subsets of .
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 ) | 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 ) | 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 ) | 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 |