Definition:Topologically Distinguishable
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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.
Topologically Indistinguishable
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:
- $\forall U \in \tau: x \in U \iff y \in U$
Also see
- Results about topological distinguishability can be found here.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): topologically distinguishable