Symmetric space: Difference between revisions

Latest revision as of 15:53, 28 January 2012

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

Definition

A topological space $X$ is termed symmetric if it satisfies the following equivalent conditions:

Its Kolmogorov quotient is a T1 space.
There is no subspace of the space that is a Sierpinski space with the subspace topology.
Given any two topologically distinguishable points $a,b\in X$ , there exists an open subset $U$ of $X$ such that $a\in U,b\notin U$ .
given points $a,b\in X$ $a,b\in X$ , the following are equivalent:
- There exists an open subset of $X$ containing $a$ but not $b$
- There exists an open subset of $X$ containing $b$ but not $a$

Relation with other properties

Stronger properties

Property	Meaning	Intermediate notions
homogeneous space	given any two distinct points, there is a self-homeomorphism of the space sending one to the other	\|FULL LIST, MORE INFO
T1 space	all points are closed	\|FULL LIST, MORE INFO
preregular space	topologically distinguishable points can be separated by pairwise disjoint open subsets	\|FULL LIST, MORE INFO
Hausdorff space		Preregular space\|FULL LIST, MORE INFO

@@ Line 3: / Line 3: @@
 ==Definition==
-===Definition with symbols===
+A [[topological space]] <math>X</math> is termed '''symmetric''' if it satisfies the following equivalent conditions:
-A [[topological space]] <math>X</math> is termed '''symmetric''' if, given points <math>a,b \in X</math>, the following are equivalent:
+# Its [[defining ingredient::Kolmogorov quotient]] is a [[defining ingredient::T1 space]].
+# There is no subspace of the space that is a [[defining ingredient::Sierpinski space]] with the [[subspace topology]].
-* There exists an [[open subset]] of <math>X</math> containing <math>a</math> but not <math>b</math>
+# Given any two [[defining ingredient::topologically distinguishable points]] <math>a,b \in X</math>, there exists an open subset <math>U</math> of <math>X</math> such that <math>a \in U, b \notin U</math>.
-* There exists an open subset of <math>X</math> containing <math>b</math> but not <math>a</math>
+# given points <math>a,b \in X</math>, the following are equivalent:
+#* There exists an [[open subset]] of <math>X</math> containing <math>a</math> but not <math>b</math>
+#* There exists an open subset of <math>X</math> containing <math>b</math> but not <math>a</math>
 ==Relation with other properties==
@@ Line 14: / Line 16: @@
 ===Stronger properties===
-* [[Homogeneous space]]
+{| class="sortable" border="1"
-* [[T1 space]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+|[[Weaker than::homogeneous space]] || given any two distinct points, there is a self-homeomorphism of the space sending one to the other || || ||{{intermediate notions short|symmetric space|homogeneous space}}
+|-
+| [[Weaker than::T1 space]] || all points are closed || || || {{intermediate notions short|symmetric space|T1 space}}
+|-
+| [[Weaker than::preregular space]] || topologically distinguishable points can be separated by pairwise disjoint open subsets || || || {{intermediate notions short|symmetric space|preregular space}}
+|-
+| [[Weaker than::Hausdorff space]] || || || || {{intermediate notions short|symmetric space|Hausdorff space}}
+|}