# Definition:Topologically Distinguishable

## Definition

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

Let $x, y \in X$.

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

That is, either:

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

or:

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

or both.

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

### Topologically Indistinguishable

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

That is:

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