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

## TL;DR

This paper explores the order topology on orthocomplemented posets of linear subspaces in pre-Hilbert spaces, linking topological properties to the completeness of the space and implications for quantum logic models.

## Contribution

It introduces a topological framework for pre-Hilbert space logics and characterizes space completeness via separation properties of the order topology.

## Key findings

- Completeness of pre-Hilbert space characterized by separation properties.
- Lack of probability theory on incomplete pre-Hilbert space logics explained topologically.
- Order topology provides insights into the structure of quantum logic models.

## Abstract

Motivated by the Hilbert-space model for quantum mechanics, we define a pre-Hilbert space logic to be a pair $(S,\el)$, where $S$ is a pre-Hilbert space and $\el$ is an orthocomplemented poset of orthogonally closed linear subspaces of $S$, closed w.r.t. finite dimensional perturbations, (i.e. if $M\in\el$ and $F$ is a finite dimensional linear subspace of $S$, then $M+F\in \el$). We study the order topology $\tau_o(\el)$ on $\el$ and show that completeness of $S$ can by characterized by the separation properties of the topological space $(\el,\tau_o(\el))$. 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.