$\mathbb R_l$ is not connected.

Question

How to show $\mathbb R_l$ (lower limit topology on $\mathbb R$) is not connected?Means how any basis element of $\mathbb R_l$ can be written as the union of two separated sets?

Answer 1

Any open set $[a,b)$ can be written as $[a,c) \cup [c,b)$ for $c$ between $a$ and $b$, so all open sets can be decomposed into non-empty, disjoint open sets.

Answer 2

The sets $[0,\infty)$ and $(-\infty,0)$ are both open in $\Bbb R_l$.