Transition from local to global of dFHE and dWCHP

TL;DR

This paper investigates the transition from local to global properties of directed fiber homotopy equivalences and directed weak covering homotopy properties in directed topology, establishing key theorems and introducing new concepts.

Contribution

It proves Dold type theorems and a tom Dieck-Kamps-Puppe type theorem for directed topology, and introduces new notions like d-halo and d-SEP.

Findings

Established theorems connecting local and global properties in directed topology.

Introduced new concepts such as d-halo and d-SEP.

Provided examples and counterexamples illustrating the concepts.

Abstract

In a previous paper [22] the author studied the directed weak covering homotopy property (dWCHP)and directed weak fibrations in the category dTop of directed spaces in the sense of M. Grandis [12], [13], [14]. This type of maps extend to the category dTop the well known Dold's (or weak)fibrations [6]. In this paper the transition from local to global of the dFHE (directed fiber homotopy equivalence) and the dWCHP are studied by proving two Dold type theorems and respectively a tom Dieck-Kamps-Puppe type theorem [3]. Some new notions of directed topology are defined: d-halo, d-SEP, d-numerable covering, d-shrinkable. Some examples and counterexamples are given.