Definition:Separated by Neighborhoods/Points/Open Sets

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Definition

Let $T = \struct {S, \tau}$ be a topological space.


Let $x, y \in S$ such that:

$\exists U, V \in \tau: x \in U, y \in V: U \cap V = \O$


That is, that $x$ and $y$ both have open neighborhoods in $T$ which are disjoint.


Then $x$ and $y$ are described as separated by (open) neighborhoods.


Thus two points $x$ and $y$ are separated by neighborhoods if and only if the two singleton sets $\set x$ and $\set y$ are separated by open neighborhoods as sets.