A theorem proving the irreversibility of the biological arrow of time, based on fixed points in the brain as a compact, Δ-complete topological space Available to Purchase

A physical space can exist as a collection of closed topologies in the intersections of abstract topological subspaces provided with non-equal dimensions. Furthermore, the ordered sequence of mappings of one to another intersection provides an arrow of time which is shared by all connected systems of closed, involving those of the brain type with other types (i.e., physical objects of all categories). The topology of closed spaces associates fixed points of the Brouwer’s type with fixed points of the Banach’s type. The former are specific of each closed and the latter drive the information from the outside space to mental images inside a closed, through mappings of Jordan’s points. The set of fixed points thus provides the properties of both perception and self in living organisms. Conditions for existence of various kinds of Banach’s type fixed points are fulfilled by the mathematical brain, since it is both a discrete finite structure, thus a compact topological space, and provided with a set distance (Δ), thus Δ-complete. Finally, since (i) iterates in a sequence of mappings include at least a surjective component and (ii) not identical (if even existing) fixed points would be generated by the non-surjective property which would characterize reciprocal mappings, in either metric or nonmetric setting, the reversion of biological time would break the direct link of the self with perception functions. Thus, while time could be reversible for physics, it is perceived as irreversible for biology, although physical and biological objects share a common space.