Order topology on orthocomplemented posets of linear subspaces of a pre-Hilbert space

Motivated by the Hilbert-space model for quantum mechanics, we define a pre-Hilbert space logic to be a pair $(S,\text{el})$, where $S$ is a pre-Hilbert space and $\text{el}$ is an orthocomplemented poset of orthogonally closed linear subspaces of $S$, closed w.r.t. finite dimensional perturbations, (i.e. if $M\in \text{el}$ and $F$ is a finite dimensional linear subspace of $S$, then $M+F\in \el$)\mathrm{.}Westudytheordertopology$\tau_o(\el)$on$\el$ and show that completeness of $S$ can by characterized by the separation properties of the topological space $(\text{el},{\tau }_o(\text{el}))$. It will be seen that the remarkable lack of a proper probability-theory on pre-Hilbert space logics – for an incomplete $S$ – comes out elementarily from this topological characterization.