Definition:Separated

Definition

Sets

Let $\left({X, \vartheta}\right)$ be a topological space.

Let $A, B \subseteq X$ such that:

$A^- \cap B = A \cap B^- = \varnothing$

where $A^-$ denotes the closure of $A$ in $X$.

Then $A$ and $B$ are described as separated.

Points

Let $\left({X, \vartheta}\right)$ be a topological space.

Let $x, y \in X$ such that both of the following hold:

• $\exists U \in \vartheta: x \in U, y \notin U$
• $\exists V \in \vartheta: y \in V, x \notin V$

Then $x$ and $y$ are described as separated.

Also see

• Results about separated sets can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Closures and Interiors