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Suppose $\mathbb{Q}$ is endowed with the subspace topology of$\mathbb{R}$ Does it follow that $\mathbb{Q}$ is connected?

MY attempt: We can sue fact that $\mathbb{Q}$ is countable and so $\mathbb{Q} = \bigcup \{x\} $. And singletons are connected. Can we conclude that the union is connected?

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No.$\ \ \ \ \ $ – bof Oct 29 '13 at 1:04
up vote 3 down vote accepted

In the relative topology, $\mathbb{Q}$ is totally disconnected.

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some might say totally disconnected. – Dan Rust Oct 29 '13 at 1:07
Yerright. I meant "totally" in the sense that each point is a connected component. – ncmathsadist Oct 29 '13 at 1:13

Why would the union of two connected sets be connected? Think about why this is not true in general. The rationals are certainly not connected.

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Nor is a finite set with more than two elements. – ncmathsadist Oct 29 '13 at 1:18
Not necessarily. The indiscreet topology on the set of 2 points is certainly connected. – LASV Oct 29 '13 at 1:19
Not as as subset of the line; that is the context I am responding to. – ncmathsadist Oct 29 '13 at 1:21
Ah. I apologise. – LASV Oct 29 '13 at 1:21
no worries here. – ncmathsadist Oct 29 '13 at 1:24

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