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. Given any two topologically distinguishable points $a,b\in X$, there exists an open subset $U$ of $X$ such that $a\in U,b\notin U$.
3. 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

• Homogeneous space
• T1 space