Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
Look here. –  k.stm Mar 28 '13 at 13:21

1 Answer 1

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.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.