Definition:Separated by Neighborhoods/Points

Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

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.

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 $\left\{{x}\right\}$ and $\left\{{y}\right\}$ are separated by neighborhoods as sets.