Discrete model of spacetime in terms of inverse spectra of the $T_0$ Alexandroff topological spaces

The theory of inverse spectra of $T_0$ Alexandroff topological spaces is used to construct a model of $T_0$-discrete four-dimensional spacetime. The universe evolution is interpreted in terms of a sequence of topology changes in the set of $T_0$-discrete spaces realized as nerves of the canonical partitions of three-dimensional compact manifolds. The cosmological time arrow arises being connected with the refinement of the canonical partitions, and it is defined by the action of homomorphisms in the proper inverse spectrum of three-dimensional $T_0$-discrete spaces. A new causal order relation in this spectrum is postulated having the basic properties of the causal order in the pseudo-Riemannian spacetime however also bearing certain quasi-quantum features. An attempt is made to describe topological changes between compact manifolds in terms of bifurcations of proper inverse spectra; this led us to the concept of bispectrum. As a generalization of this concept, inverse multispectra and superspectrum are introduced. The last one enables us to introduce the discrete superspace, a discrete counterpart of the Wheeler--DeWitt superspace.