Is $[a,b]$ in ${\mathbb{R}_L}$ connected in the subspace topology?
I am trying to see whether or not the IVT applies for $[a,b]$ in the topology inherited from $\mathbb{R}_L$ instead of $[a,b]$ inherited from $\mathbb{R}_U$. My proof of the IVT relies on $[a,b]$ being connected in the subspace topology so this would be helpful.
For a counterexample to the intermediate value theorem when $[a,b]$ has the Sorgenfrey (or lower limit) topology, fix a point $c\in(a,b)$, and define $f:[a,b]\to\Bbb R$ as follows:
$$f(x)=\begin{cases} 0,&\text{if }a\le x<c\\ 1,&\text{if }c\le x\le b\,. \end{cases}$$
Because $[a,c)$ is both open and closed in $[a,b]$, $f$ is continuous, and $f$ assumes no value strictly between $f(a)=0$ and $f(b)=1$.