# Prove that the Sorgenfrey line is not connected

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?

• Looks ok to me...
– 5xum
Nov 23 '16 at 9:13

Another way is to show that $\mathbb R = (-\infty, 0)\cup [0,\infty)$ is one way of splitting $\mathbb R$ into two open sets.