Definition:Separated by Neighborhoods

Definition

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

Sets

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

$\exists N_A, N_B \subseteq S: \exists U, V \in \tau: A \subseteq U \subseteq N_A, B \subseteq V \subseteq N_B: N_A \cap N_B = \O$

That is, that $A$ and $B$ both have neighborhoods in $T$ which are disjoint.

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

Points

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

$\exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \in U \subseteq N_x, y \in V \subseteq N_y: N_x \cap N_y = \O$

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

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