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?


[Non-isomorphic projective planes on $\omega$](https://mathoverflow.net/questions/261436/non-isomorphic-projective-planes-on-omega)



Continuous projective geometry on the interval
If $(P,L)$ is an abstract projective plane, then for any point $p\in P$ and any line $\ell\in L$ not incident to $p$ there is a bijection between the set of points incident to $\ell$ and the set of lines incident to $p$. Under a reasonable definition of "topological projective plane" this bijection should be a homeomorphism.


If $P=[0,1]$, then the space of all lines incident to $0$ has a continuous surjection from the connected space $(0,1]$. It follows that for each line $\ell$ the space of all points incident to $\ell$ is connected. But one of these lines has both $0$ and $1$ in it, and the only connected subset of an interval containing both endpoints is the whole interval. Contradiction.


