About no differentiable optimization

The subject matter of the present thesis is generally an elaboration of the implications resulted from the
adoption of the definition of an almost continuous map as it appeared lately in the literature. At the onset, the relationships to continuous, almost continuous, weakly continuous and 0-continuous maps are discussed and then illustrated by examples. Stepping forward in this development, the topologically important functional processes, restriction, extension, composition and cross product were studied within the context of almost continuous maps as the relevant maps. A special example of such maps, the quotient map, was given special attention. Next, almost continuous maps were sought for effects on special types of spaces, namely a Hausdorff, a compact, a locally compact and a connected space. A point that was carefully examined is whether an almost continuous preserves these properties upon mapping its range. It was found specifically that almost continuous maps are connectivity preserving. Two important theorems involving a Orison's space and an almost continuous map are presented. Another theorem concerning the retraction of an almost continuous map was demonstrated. A concluding account of the relationships with closed graphs and graph mappings was also presented. The special case of a closed graph whose range is a locally compact Hausdorff space was specifically studied.