Is the space $\Bbb{R}_l$ connected?

Question

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.

Answer 1

The equality $[0,2) = [0,1) \cup [1,2)$ proves that $[0,2)$ is not connected.

Hint:

Show that $(-\infty,0)$ and $[0,\infty)$ are open, writing them as union of sets from the basis.

Answer 2

$(-\infty,0) = \bigcup_{n \in \Bbb{N}}[-n,-n+1)$ and $[0,\infty) = \bigcup_{k = 0}^\infty [k,k+1)$ are two disjoint open sets separating $\Bbb{R}_l$, so $\Bbb{R}_l$ is disconnected.