Base-base paracompactness and subsets of the Sorgenfrey line

summary:A topological space $X$ is called base-base paracompact (John E. Porter) if it has an open base $\mathcal B$ such that every base ${\mathcal B' \subseteq \mathcal B}$ has a locally finite subcover $\mathcal C \subseteq \mathcal B'$. It is not known if every paracompact space is base-base paracompact. We study subspaces of the Sorgenfrey line (e.g. the irrationals, a Bernstein set) as a possible counterexample