Kompaktnost v metrických prostorech a její aplikace

Abstract

This thesis deals with the concept of compactness in metric spaces. In the introductory part, the necessary basic notions and various equivalent characterizations of compactness are presented, in particular the definition via open covers and via subsequences. The next part of the thesis is devoted to the consequences of compactness, including, for example, the Weierstrass theorem on the existence of extrema of continuous functions and the Heine–Cantor theorem on uniform continuity. Furthermore, criteria for compactness in specific metric spaces, such as spaces of continuous functions and sequence spaces, are introduced and discussed. The final part of the thesis focuses on selected applications of compactness, for example in the theory of differential equations or in connection with compact operators. The aim of the thesis is to provide a comprehensive overview of the concept of compactness, its various characterizations, and its importance in different areas of mathematical analysis.