Definition:Separated Sets
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Definition
Let $T = \struct {S, \tau}$ 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 $U \cap B = \O$
- $B \subset V$ and $V \cap A = \O$
where $\O$ 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
- Definition:Separation (Topology), a definition which is linked by Components of Separation are Separated Sets.
- Definition:Separable Space, an unrelated definition.
- Definition:Tychonoff Separation Axioms, a classification system for topological spaces.
- Results about separated sets can be found here.