Symmetric space: Difference between revisions
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# Its [[defining ingredient::Kolmogorov quotient]] is a [[defining ingredient::T1 space]]. | # 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>. | # There is no subspace of the space that is a [[defining ingredient::Sierpinski space]] with the [[subspace topology]]. | ||
# 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>. | |||
# 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> | ||
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 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 , 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
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | 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 |