for metric space, we can say a point must be isolated point or cluster point. In generally, why it is not true? also, it is known that, a set is closed iff it is contained all cluster point. can we say, a set is closed iff it has no isolated point?
a point of a subset in any topology either isolated or cluster point?
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general-topology
1 Answers
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No, we can’t say that a set is closed iff it has no isolated: in a $T_1$ space every finite set is closed, and every point of a finite set is isolated in that set. On the other hand, the set $\Bbb Q$ of rationals has no isolated point, but it is not closed in $\Bbb R$.
It is true, however, that in any space every point of a set $A$ is either an isolated point of $A$ or a cluster point of $A$, simply because there is no third possibility.
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0@ege: It depends on the context. In and of itself it has no particular significance; it’s just one extreme possibility. At the other extreme are so-called *scattered* spaces: $X$ is scattered if every subset of $X$ has an isolated point in its relative topology. – 2012-12-12