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In a topological space $X$, quoted from Wikipedia:

A point $x ∈ X$ is a cluster point of a sequence $(x_n)_{n ∈ N}$ if, for every neighbourhood $V$ of $x$, there are infinitely many natural numbers $n$ such that $x_n ∈ V$. If the space is sequential, this is equivalent to the assertion that $x$ is a limit of some subsequence of the sequence $(x_n)_{n ∈ N}$.

I was wondering when a topological space is not necessarily sequential, what is the relation between cluster point of a sequence and limit point of some subsequence of the sequence?

When the topological space is sequential space, why are the two equivalent?

Thanks and regards!

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    Wikipedia is wrong: we need ($T_1$) Fréchet-Urysohn spaces for this to hold, also for the statement that there is a sequence from $S \setminus \{ p \}$ converging to $p$ if $p$ is a limit point of $S$. See my answer below.2012-02-01

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