Topology and statistics in zero dimensions
The author studies finite topological \(T_ 0\)-spaces \(X\) as models for space-times. \textit{R. D. Sorkin} [Int. J. Theor. Phys. 30, No. 7, 923-947 (1991; Zbl 0733.54001)] noted that every such space carries a natural partial order denoted by ``\(\to\)'': For \(x,y \in X\), \(x \to y\) if \(x \in \Lambda (y)\) where \(\Lambda (y)\) is the intersection of all open sets containing \(y\). Using this fact it is shown how the fundamental group of the topological space \(X\) can be determined from the corresponding Hasse diagram of the partially ordered set \(X\). The last section is devoted to the study of statistics on the configuration space \((X \times X - X)/Z_ 2\) of \(X\). The idea to introduce finite topological spaces as (local) models for space-time seems to be very promising. It complements in a certain sense past contributions concerning topological spaces with a ``fundamental length'' (see, as just one example, \textit{S. P. Gudder} [SIAM J. Appl. Math. 16, 1011-1019 (1968; Zbl 0179.302)]).
scientific article; zbMATH DE number 575506