Quasi-shape theory of locally finite and paracompact spaces

TL;DR

This paper introduces quasi-shape theory, a new approach that effectively captures the weak homotopy type of locally finite spaces and aligns with traditional shape theory for paracompact spaces.

Contribution

It proposes a novel variant of shape theory called quasi-shape, bridging the gap between shape theory and weak homotopy types for different classes of spaces.

Findings

Quasi-shape is isomorphic to the weak homotopy type for locally finite spaces.

Quasi-shape is tural-equivalent to the ordinary shape for paracompact spaces.

The theory extends shape concepts to spaces with no separation axioms.

Abstract

Shape theory works nice for (Hausdorff) paracompact spaces, but for spaces with no separation axioms, it seems to be quite poor. However, for finite and locally finite spaces their weak homotopy type is rather rich, and is equivalent to the weak homotopy type of finite and locally finite polynedra, respectively. In the paper there is proposed a variant of shape theory called quasi-shape, which suits both paracompact and locally finite spaces, i.e. the quas-shape is isomorphic to the weak homotopy type for locally finite spaces, and is \natural-equivalent to the ordinary shape in the case of paracompact spaces.