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Problem:

Let $ \mathbb{R}_l $ denote the topological space whose underlying set is the real line $ \mathbb{R} $ and the topology is generated by the half closed intervals $ [a,b) $. Prove that the topological space $ \mathbb{R}_l $ is not connected.

My proof:

By definition, a connected space has no non-empty proper subset that are both open and closed. Any half-closed set $ [a,b) $ is by definition open in $ \mathbb{R}_l $, but every set $ [a,b) $ is also closed since its compliment $$\mathbb{R}_l\backslash[a,b)=(-\infty,a)\cup [b,\infty)=\bigcup_{\substack{n=1}}^\infty [-n,a)\cup [b,\infty)$$ is open. It follows that the topological space $ \mathbb{R}_l $ is not connected.

Question: Is my proof correct?

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    $\begingroup$ Looks ok to me... $\endgroup$
    – 5xum
    Nov 23 '16 at 9:13
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Looks fine to me.

Another way is to show that $\mathbb R = (-\infty, 0)\cup [0,\infty)$ is one way of splitting $\mathbb R$ into two open sets.

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