Nonstandard homology theory for uniform spaces

TL;DR

This paper introduces a novel homology theory for uniform spaces called μ-homology, based on hyperfinite chains, satisfying Eilenberg-Steenrod axioms and characterizing chain-connectedness.

Contribution

It develops μ-homology theory for uniform spaces, proving it satisfies axioms, characterizes chain-connectedness, and introduces S-homotopy, extending homology concepts.

Findings

μ-homology satisfies Eilenberg-Steenrod axioms

Characterizes chain-connectedness via μ-homology

X and dense subsets share the same μ-homology

Abstract

We introduce a new homology theory of uniform spaces, provisionally called $\mu$-homology theory. Our homology theory is based on hyperfinite chains of microsimplices. This idea is due to McCord. We prove that $\mu$-homology theory satisfies the Eilenberg-Steenrod axioms. The characterization of chain-connectedness in terms of $\mu$-homology is provided. We also introduce the notion of S-homotopy, which is weaker than uniform homotopy. We prove that $\mu$-homology theory satisfies the S-homotopy axiom, and that every uniform space can be S-deformation retracted to a dense subset. It follows that for every uniform space $X$ and any dense subset $A$ of $X$, $X$ and $A$ have the same $\mu$-homology. We briefly discuss the difference and similarity between $\mu$-homology and McCord homology.