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?
Best Answer
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.