Can someone provide an example of a locally connected Hausdorff space not consisting of a single point?
Provide an example of a locally connected Hausdorff space not consisting of a single point.
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general-topology
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1$\Bbb R$. $[0,1]$, if you want it to be compact. – 2012-04-14
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0i guess i'm having trouble of what can and what cannot be interpreted as a point. – 2012-04-14
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0Both 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 $\{x\}$. – 2012-04-14
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1"Not consisting of a single point" means the space itself is not a single point, not that it doesn't have points. – 2012-04-14
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0yes, i assumed it could't contain any. sorry. thank you** – 2012-04-14
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0Hi @BrianM.Scott, I am trying to understand your commend. What is $\mathbb{R}.[0,1]$? – 2013-07-28
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0@WishingFish ${\Bbb R}$ is the space of real numbers. A second example given by Brian, which is compact, is the closed interval $[0,1]$ which is defined as the set of reals $x$ satisfying $0\le x\le 1$, with the subspace topology. – 2013-07-28
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0Oh that's what the very abbreviated linguist meant! Thank you @anon! I was just wondering where have you been..... – 2013-07-28
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0@WishingFish: As anon said, two separate examples. One example is $\Bbb R$. If you want a compact example, you can use $[0,1]$ instead. – 2013-07-28
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0Yes, got it. Glad to see your nice answer, thanks @BrianM.Scott – 2013-07-28
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0@WishingFish: You’re welcome. – 2013-07-28
1 Answers
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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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0Fun, accurate, and angry answer. Like it. – 2013-07-28