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

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$\Bbb R$. $[0,1]$, 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
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"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

1 Answer 1

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!

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Fun, accurate, and angry answer. Like it. –  WishingFish Jul 28 '13 at 3:50

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