Definition:Topologically Distinguishable

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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:

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