Mapping Spaces and Postnikov Invariants

If $q:Y\longrightarrow{B}$ is a fibration and $Z$ is a space, then the free range mapping space $Y!Z$ has a collection of partial maps from $Y$ to $Z$ as underline space, i.e. those such maps whose domains are individual fibre of $q$. It is shown in \cite{P. Booth6} that these spaces have applications to several topics in homotopy theory. These such results are given in complete detail, concerning identification, cofibrations and sectioned fibrations. The necessary topological foundations for two none complicated applications, i.e. to the cohomology of fibrations and the classification of Moore-Postinikov systems, are given, and the applications themselves outlined. The usual argument is in the context of the usually category of all topological spaces, and this necessarily introduces some new problems. Whenever we work with exponential laws for mapping spaces, in that category, we will usually find that we are forced to assume that some of the spaces are locally compact and Hausdorff, which detracts considerable from the generality of the results obtained. In this paper we develop the corresponding theory in the category of compactly generated or k-spaces, which is free of the inconvenient assumptions referred to above. In particular, we obtain the k-space version of the applications to identifications, cofibrations and sectioned fibrations, and establish improved foundations for the k-versions of the other two applications, i.e. the cohomology of fibrations and a classification theory for Moore-Postnikov factorizations.