  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.
