Theory of Compact Hausdorff Shape

TL;DR

This paper introduces a new shape theory called compact Hausdorff shape (CH-shape) for Hausdorff spaces, using an internal method and direct system approach, and develops its homology theory.

Contribution

It establishes the CH-shape theory for Hausdorff spaces, preserving key properties of H-shape and extending homology theory to this new framework.

Findings

CH-shape theory successfully developed for Hausdorff spaces

Homology theory for CH-shape established with exactness and duality

Preserves most properties of existing H-shape theory

Abstract

In this paper, we aim to establish a new shape theory, compact Hausdorff shape (CH-shape) for general Hausdorff spaces. We use the "internal" method and direct system approach on the homotopy category of compact Hausdorff spaces. Such a construction can preserve most good properties of H-shape given by Rubin and Sanders. Most importantly, we can moreover develop the entire homology theory for CH-shape, including the exactness, dual to the consequence of Marde\v{s}i\'c and Segal.