Definition:Topologically Distinguishable

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

Let $x, y \in S$.

Then $x$ and $y$ are topologically distinguishable they do not have exactly the same neighborhoods.

That is, either:
 * $\exists U \in \tau: x \in U \subseteq N_x \subseteq S: y \notin N_x$

or:
 * $\exists V \in \tau: y \in V \subseteq N_y \subseteq S: x \notin N_y$

or both.

That is, at least one of the elements $x$ and $y$ has a neighborhood that is not a neighborhood of the other.