One-point reductions of finite spaces,h–regular CW–complexes and collapsibility

Abstract

We investigate one-point reduction methods of finite topological spaces. These methods allow one to study homotopy theory of cell complexes by means of elementary moves of their finite models. We also introduce the notion of h-regular CW-complex, generalizing the concept of regular CW-complex, and prove that the h-regular CW-complexes, which are a sort of combinatorial-up-to-homotopy objects, are modeled (up to homotopy) by their associated finite spaces. This is accomplished by generalizing a classical result of McCord on simplicial complexes. 55U05, 55P15, 57Q05, 57Q10; 06A06, 52B70

Paths, Homotopy and Reduction in Digital Images

“…The following definition of a wsimple point ("w" stands for "weak") and their properties are due to Barmak and Minian [4] who call them γ-points. Bertrand in [6] defines a quite similar notion.…”

Inverse limits of finite topological spaces

“…There has recently been renewed interest in finite topological spaces; for example, in [2], Barmak and Minian present an approach to simple homotopy theory which is based on McCord's correspondence. In particular, they introduce the notion of a collapse of finite spaces and prove that it corresponds under this association to simplicial collapse, while in [1], they introduce a broader class of spaces than simplicial complexes, namely the so-called "h-regular CW complexes", to which McCord's analysis and their extension apply.…”

Topology on Digital Label Images

“…The following definition of γ-simple points 8 and their properties are due to Barmak and Minian [42].…”