nLab symmetric topological space

Definition

A topological space $(X,\tau)$ is called a symmetric topological space or an $R_0$-topological space if for all elements $x \in X$ and $y \in X$, if for all open sets $U \in \tau$, $x \in U$ implies that $y \in U$, then for all open sets $U \in \tau$, $y \in U$ implies that $x \in U$. Or equivalently, if its specialization preorder is an equivalence relation.

Properties

A $T_1$-topological space is a symmetric topological space which is also a Kolmogorov topological space. The quotient of a symmetric topological space $(X, \tau)$ by its specialization order equivalence relation is a $T_1$-topological space, the Kolmogorov quotient of $X$.

 Examples

Every set $X$ with its power set $\mathcal{P}(X)$ is a symmetric topological space with respect to the specialization order equivalence relation defined as

$$x \equiv y \coloneqq \forall P \in \mathcal{P}(X).P(x) \iff P(y)$$

for all $x \in X$ and $y \in X$.

Related concepts

• specialization order
• separation axiom

preorder	partial order	equivalence relation	equality
topological space	Kolmogorov topological space	symmetric topological space	accessible topological space