Definition:Separated Sets
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Definition
Let $T = \left({S, \tau}\right)$ be a topological space.
Let $A, B \subseteq S$.
Definition 1
$A$ and $B$ are separated (in $T$) if and only if:
- $A^- \cap B = A \cap B^- = \O$
where:
Definition 2
$A$ and $B$ are separated (in $T$) if and only if there exist $U,V\in\tau$ with:
- $A\subset U$ and $U\cap B = \varnothing$
- $B\subset V$ and $V\cap A = \varnothing$
where $\varnothing$ denotes the empty set.
$A$ and $B$ are said to be separated sets (of $T$).
Also known as
When $A$ and $B$ are separated in $T$, they are said to separate $T$.
Also see
- Equivalence of Definitions of Separated Sets
- Results about separated sets can be found here.