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

Can someone provide an example of a locally connected Hausdorff space not consisting of a single point?

share|cite|improve this question
$\Bbb R$. $[0,1]$, if you want it to be compact. – Brian M. Scott Apr 14 '12 at 0:38
i guess i'm having trouble of what can and what cannot be interpreted as a point. – The Substitute Apr 14 '12 at 0:39
Both of the examples that I gave have $2^\omega=\mathfrak{c}$ points. The one-point space is just a singleton set $\{x\}$ and the open sets $\varnothing$ and $\{x\}$. – Brian M. Scott Apr 14 '12 at 0:39
"Not consisting of a single point" means the space itself is not a single point, not that it doesn't have points. – anon Apr 14 '12 at 0:41
yes, i assumed it could't contain any. sorry. thank you** – The Substitute Apr 14 '12 at 0:42
up vote 4 down vote accepted

It is true that in English "not consisting of a single X" could be interpreted as not having any Xs.

However, here we mean the space is not itself a single point (see Brian's comment about the one-point space). It does not mean the space doesn't have any points!

share|cite|improve this answer
Fun, accurate, and angry answer. Like it. – WishingFish Jul 28 '13 at 3:50

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.