Relationship among various Vietoris-type and microsimplicial homology theories

TL;DR

This paper explores the relationships among various Vietoris-type and microsimplicial homology theories, establishing isomorphisms under specific conditions and demonstrating shared properties, thereby clarifying their connections in topology.

Contribution

It proves isomorphisms between McCord's, Korppi's, and the authors' homology theories for specific classes of spaces, and relates these to classical Vietoris homology.

Findings

McCord's and our homology are isomorphic for all compact uniform spaces.

Korppi's and our homology are isomorphic for all fine uniform spaces.

Our homology is isomorphic to uniform Vietoris homology for precompact uniform spaces.

Abstract

In this paper, we clarify the relationship among the Vietoris-type homology theories and the microsimplicial homology theories, where the latter are nonstandard homology theories defined by M. C. McCord (for topological spaces), T. Korppi (for completely regular topological spaces) and the author (for uniform spaces). We show that McCord's and our homology are isomorphic for all compact uniform spaces and that Korppi's and our homology are isomorphic for all fine uniform spaces. Our homology shares many good properties with Korppi's homology. As an example, we outline a proof of the continuity of our homology with respect to uniform resolutions. S. Garavaglia proved that McCord's homology is isomorphic to Vietoris homology for all compact topological spaces. Inspired by this result, we prove that our homology is isomorphic to uniform Vietoris homology for all precompact uniform spaces and that Korppi's homology is isomorphic to normal Vietoris homology for all pseudocompact completely regular topological spaces.