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Let ($X, T $) be a topological space where $X$ is a uncountable space and $T$ is the discrete topology . Then it is given that $x$ is not a limit point of the sequence $x, x,$ . . . considered as a set, although it is an adherent point of the set.

But I could not find the proof. Can anyone help me please?

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Look here. – k.stm Mar 28 '13 at 13:21

In the discrete topology, every set and particularly every singleton is open. The singleton $\{x\}$ is an open neighbourhood of $x$, making $x$ an isolated point.

A more general statement holds: No isolated point is the limit point of any set. This is described and proven here. (It’s really just the definitions of limit point vs. isolated point.)

Oh, and for $x$ to be a limit point of a sequence $(x_n)_{n ∈ ℕ}$, by definition it needs to be a limit point of the set $\{x_n;\, n ∈ ℕ\}$.

The point $x$ is still the limit of the constant sequence $(x,x,…)$ just as $0$ would be the limit, but no limit point of the constant sequence $(0,0,…)$ in $ℝ$ with the usual topolgy.

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