# Totally disconnect space and the real number set with lower limit topology

Is $\mathbb R_l$ totally disconnected? We know $\mathbb R_l$ is finer than $\mathbb R$ and intervals and one point sets are only connected subsets of $\mathbb R$ .hence only possible connected sets in $\mathbb R_l$ is intervals and one point set .as intervals are seperated by $(-\infty ,a),[a,\infty)$ in $\mathbb R_l$. Therefore one point sets are only connected sets in $\mathbb R_l$.

Am I wrong?

• It seems fine to me. – pisco Nov 12 '17 at 5:29
• Correct. The Sorgenfrey line has a base (basis) of open-and-closed sets. – DanielWainfleet Nov 12 '17 at 10:57