Symmetric space: Difference between revisions

Revision as of 15:39, 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.
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 $a,b\in X$ , 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

@@ 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]].
+# Given any two [[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>.
+# 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>