the current ebook is the end result of efforts to introduce topological connectedness as one of many uncomplicated instruments for the learn of important stipulations for an extremum. it appears this monograph is the 1st booklet within the conception of maxima and minima the place topological connectedness is used so generally for this function. Its software allows us to procure new ends up in this sphere and to think about the classical effects from a nonstandard viewpoint. concerning the sort of the current booklet it's going to be remarked that it truly is relatively straightforward. the writer has made consistent efforts to make the ebook as self-contained as attainable. definitely, familiarity with the fundamental evidence of topology, useful research, and the idea of optimization is believed. The booklet is written for utilized mathematicians and graduate scholars drawn to the speculation of optimization and its functions. We current the synthesis of the well-known Dybovitskii'-Milyutin ap­ proach for the research of precious stipulations for an extremum, in line with sensible research, and topological equipment. This synthesis permits us to teach that during a few instances we now have the subsequent vital outcome: if the Euler equation has no non trivial resolution at some degree of an extremum, then a few inclusion is legitimate for the functionals belonging to the twin house. This common result's bought for an optimization challenge thought of in a lin­ ear topological house. We additionally convey an program of our consequence to a couple difficulties of nonlinear programming and optimum control.