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## Main Question or Discussion Point

A set S is closed iff it contains all its adherent points iff it contains all its accumulation point?

From what I know, in general accumulation point is a subset of adherent point, but if supposed I have a closed set, then the "if and only if" forces me to conclude that accumulation point = adherent points.

Supposed they are not equal, then since having all accumulation points inside the set already makes it a closed set, then a closed set doesn't necessarily contains all its adherent points which contradicts the definition.

Sorry if this is an easy question, but then if my reasoning is wrong, please help me to correct it. Thanks.

From what I know, in general accumulation point is a subset of adherent point, but if supposed I have a closed set, then the "if and only if" forces me to conclude that accumulation point = adherent points.

Supposed they are not equal, then since having all accumulation points inside the set already makes it a closed set, then a closed set doesn't necessarily contains all its adherent points which contradicts the definition.

Sorry if this is an easy question, but then if my reasoning is wrong, please help me to correct it. Thanks.