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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.

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