Put $P=[0,1]$. Is there a compact subset $L$ of the hyper space of $P$ such that the pair $(P, L)$ satisfies  the following axioms of projective geometry. Furthermore  the obvious maps from the configuration space of $P$ to $L$ and configuration space of $L$ to $P$ would be continuous?

http://mathoverflow.net/questions/261436/non-isomorphic-projective-planes-on-omega