Is the space $\Bbb{R}_l$ connected ? Justify your answear.
$\Bbb{R}_l$ has a basis in this form: $\{[a,b): a,b \in \Bbb{R}, a<b \}$. A connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. I am not sure if I understand this properly. For example:
$[0,2) = [0,1) \cup [1,2)$
$[0,1)$ and $[1,2)$ are nonempty, disjoint and open because they are basic sets. So $\Bbb{R}_l$ is NOT connected.