Symmetric space

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:

1. Its Kolmogorov quotient is a T1 space.
2. There is no subspace of the space that is a Sierpinski space with the subspace topology.
3. Given any two topologically distinguishable points $a,b\in X$, there exists an open subset $U$ of $X$ such that $a\in U,b\notin U$.
4. 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

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