Fuzzy topology and geometric quantum formalism

Dodson–Zeeman fuzzy topology (FT) is considered as possible basis of geometric quantum formalism. In $1$-dimensional case the fundamental set of FT elements (points) $S$ is Poset, so that its elements (points) beside standard ordering relation $b\le c$, can obey also to the incomparability relation: $b\sim c$. To detail it, the normalized fuzzy weight $w(b,c)>0$ is introduced. If $X=\{x\}$ is continuous ordered $S$ subset, then $b$ coordinate relative to $X$ can be principally uncertain and $w(b,c)0$ characterizes its spread. In our formalism such fuzzy point $b(t)$ describes the evolving particle $m$, its state $\varphi(t)$ characterized by normalized density $w(x,t)$ and $w$ flow velocity $v(x,t)$, it's shown that $\varphi(x,t)$ is equivalent to the complex function combining this two parameters, and can be described as Dirac vector (ray) of complex Hilbert space. It's proved that $\varphi(x,t)$ evolves according to Dirac equation, the particle's interactions on fuzzy manifold are shown to be gauge invariant.