◀ ▲ ▶Branches / Topology / Definition: Isolated, Adherent, Limit, `$\omega$`Accumulation and Condensation Points
Definition: Isolated, Adherent, Limit, $\omega$Accumulation and Condensation Points
Let $(X,\mathcal O)$ be a topological space, let $U\subseteq X$ be its subset and let $x\in U.$
If every open set $O\in\mathcal O$ containing $x$ contains
 ... no other point of $U$, $x$ is called an isolated point of $U,$
 ... at least one point of $U$, it is called an adherent point of $U,$
 ... at least one point of $U$ distinct from $x,$ it is called a limit point of $U,$
 ... infinitely many points of $U$ distinct from $x,$ it is called an $\omega$accumulation point of $U,$
 ... uncountably many points of $U$ distinct from $x,$ it is called a condensation point of $U.$
Notes
 An adherent point is an umbrella term for isolated and limit points.
Mentioned in:
Definitions: 1 2 3 4
Proofs: 5 6
Propositions: 7
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References
Bibliography
 Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970