Nonstandard homology theory for uniform spaces

We introduce a new homology theory of uniform spaces, provisionally called μ-homology theory. Our homology theory is based on hyperfinite chains of microsimplices. This idea is due to McCord. We prove that μ-homology theory satisfies the Eilenberg–Steenrod axioms. The characterization of chain-connectedness in terms of μ-homology is provided. We also introduce the notion of S-homotopy, which is weaker than uniform homotopy. We prove that μ-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 μ-homology. We briefly discuss the difference and similarity between μ-homology and McCord homology.