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 if and only if 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.

The two points $x$ and $y$ are topologically indistinguishable if and only if they are not topologically distinguishable.

That is if and only if they do not exactly the same neighborhoods:

$\forall U \in \tau: x \in U \iff y \in U$