Definition:Separated by Neighborhoods/Points

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

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


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

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

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

Thus two points are separated by neighborhoods $x$ and $y$ the two singleton sets $\left\{{x}\right\}$ and $\left\{{y}\right\}$ are separated by neighborhoods as sets.

Generalizations

 * Definition:Sets Separated by Neighborhoods

Weaker conditions

 * Definition:Separated Points

Stronger conditions

 * Definition:Points Separated by Closed Neighborhoods