Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose that $X$ is Hausdorff and $x \in cl(C)$, where $C$ is countable discrete in $X$. Does there exist a sequence $\{x_n\}$ of $C$ converging to $x$?

share|cite|improve this question
up vote 3 down vote accepted

No. For example, let $X$ be the Stone–Čech compactification of $\Bbb{N}$, and $C$ be the embedded copy of $\Bbb{N}$. Then $C$ is countable and discrete, and its closure is all of $X$, but no sequence in $C$ converges to a point of $X \setminus C$.

share|cite|improve this answer

Hint: What does it mean for $x\in cl(C)$?

share|cite|improve this answer
Please say more. – Paul Dec 23 '12 at 12:23

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.