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.

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

I been reading for several hours and not yet found a question put in this way. Given any topological space:

  1. Does every sequance in $X$ determine a countable subset of $X$?

  2. Do the sets that belong the a topology on $X$ (the open sets) separete these sequences (given 1. is true) and therefore supply a structure for "the set $X$" turning it into a space?

share|cite|improve this question
up vote 0 down vote accepted

After a sequence of comments on the other answer, I believe Question 2 is asking:

Is a topology uniquely determined by its convergent sequences?

The answer to this question is no. For a counterexample, let $\omega_1$ be the first uncountable ordinal, and consider $\omega_1 + 1 = \omega_1 \cup \{\omega_1\}$ under the order topology. Every neighborhood of $\omega_1$ then contains infinitely many elements, thus $\omega_1$ is not an isolated point, but no sequence converges to $\omega_1$ (any countable increasing sequence of ordinals necessarily has a countable limit ordinal).

Sometimes we generalize the concept of sequences to nets, which do capture all information about the topology of a space.

share|cite|improve this answer

Question 1 No, the sequence $\{x_n=1\}_{n=1}^\infty$ in $\mathbb{R}$ is an infinite sequence finite subset of $X$. Unless you mean something else by "determine a countable infinite subset".

Question 2 I'm not entirely sure what you are asking here. You're question seems somewhat circular or trivial since the open sets of $X$ will by definition "supply a structure for "the set $X$" and turn it into a space" - the open sets form the topology of $X$. Moreover, it seems that your second question does not rely on whether or not Question 1 was true or false.

share|cite|improve this answer
I confused myself..lets just say a countable subset. – User1 Jul 26 '13 at 14:06
@Johan, what is the definition of countable to you? Countable means bijection with $\mathbb{N}$? – Sigur Jul 26 '13 at 14:14
@Johan Sequences are inherently countable (unless you are talking of some sort of transfinite sequence over the ordinals, however, I don't believe that is what you mean. – Christian Bueno Jul 26 '13 at 14:15
What Im asking in 2 is if these sequances determine subsets then the whole purpose of the topology is to seperate them by open sets – User1 Jul 26 '13 at 14:17
@Johan In what sense do you mean that a sequence "determines" a subset? A sequence in $X$ is a function $f:\mathbb{N}\to X$, and I interpreted the "determined" subset to be the image of $f$. Is this what you mean? – Christian Bueno Jul 26 '13 at 14:21

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.