Eilenberg-Steenrod homology and cohomology theories for \v{C}ech's closure spaces

TL;DR

This paper extends algebraic topology to cech's closure spaces, defining multiple singular (co)homology theories, establishing their properties, and exploring their homotopy and exact sequence structures.

Contribution

It introduces six new singular (co)homology groups for closure spaces and develops their homotopy theory and axiomatic framework, generalizing classical results.

Findings

Six distinct singular (co)homology groups for closure spaces.

Verification of Eilenberg-Steenrod axioms for three groups.

Existence of long exact sequences and Kfcnneth theorems.

Abstract

We generalize some of the fundamental results of algebraic topology from topological spaces to \v{C}ech's closure spaces, also known as pretopological spaces. Using simplicial sets and cubical sets with connections, we define three distinct singular (relative) simplicial and six distinct singular (relative) cubical (co)homology groups of closure spaces. Using acyclic models we show that the three simplicial groups have isomorphic cubical analogues among the six cubical groups. Thus, we obtain a total of six distinct singular (co)homology groups of closure spaces. Each of these is shown to have a compatible homotopy theory that depends on the choice of a product operation and an interval object. We give axioms for an Eilenberg-Steenrod (co)homology theory with respect to a product operation and an interval object. We verify these axioms for three of our six (co)homology groups. For the other three, we verify all of the axioms except excision, which remains open for future work. We also show the existence of long exact sequences of (co)homology groups coming from K\"unneth theorems and short exact (co)chain complex sequences of pairs of closure spaces.