Symmetric space: Difference between revisions
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# 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 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: | # 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>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> | #* There exists an open subset of <math>X</math> containing <math>b</math> but not <math>a</math> | ||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 15:40, 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 is termed symmetric if it satisfies the following equivalent conditions:
- Its Kolmogorov quotient is a T1 space.
- Given any two topologically distinguishable points , there exists an open subset of such that .
- given points , the following are equivalent:
- There exists an open subset of containing but not
- There exists an open subset of containing but not
