Transition from local to global of dFHE and dWCHP

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.