Any set of the form $\{x, y\}$ is disconnected. Wouldn't this imply that the rational numbers is a discrete space, since $\{x\}$ and $\{y\}$ are open?
The rational numbers are totally disconnected but not a discrete space?
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real-analysis
general-topology
connectedness
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6$\{x\}$ and $\{y\}$ are only open relative to $\{x,y\}$ but not in $\mathbb R$. – 2012-04-27
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3(nor in $\mathbb Q$) – 2012-04-27
1 Answers
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An open set of the subspace topology on $\Bbb Q$ is obtained by taking the intersection of $\Bbb Q$ with an open set of $\Bbb R$. Any non-empty open set in $\Bbb R$ contains an interval. So, could $\{x\}$ possibly be open in $\Bbb Q$?
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0Isn't {x} open in the subspace topology on {x, y}? – 2012-04-27
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0Yes, but is $\{x,y\}$ open in $\mathbb Q$? – 2012-04-27
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1@Mike $\{x\}$ is open in the space $\{x,y\}$. But singleton sets are not open in $\Bbb Q$. This is so because you cannot write $\{x\}=O\cap \Bbb Q$ with $O$ an open set of $\Bbb R$ (a non-empty open subset of $\Bbb R$ contains infinitely many rationals). – 2012-04-27
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0Okay thanks. I forgot that a subset of a space being connected/disconnected means being connected/disconnected in its subspace topology. – 2012-04-27
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0@Mike $\{x\}$ is open in $\{x\}$ but that doesn't mean that $\{x\}$ is open in $\mathbb R$ or that $\mathbb R$ is discrete. – 2016-07-23