Definition:Topologically Distinguishable

Definition

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

Let $x, y \in X$.

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

That is, either:

$\exists U \in \vartheta: x \in U \subseteq N_x \subseteq X: y \notin N_x$

or:

$\exists V \in \vartheta: y \in V \subseteq N_y \subseteq X: x \notin N_y$

or both.

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

If $x$ and $y$ are topologically distinguishable points, then the singleton sets $\left\{{x}\right\}$ and $\left\{{y}\right\}$ must be disjoint.

Topologically Indistinguishable

Two points $x$ and $y$ are topologically indistinguishable if they are not topologically distinguishable.

That is:

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