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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
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up vote 3 down vote accepted

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

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some might say totally disconnected. –  Daniel 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
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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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