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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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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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