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Diplomski rad podijeljen je na tri glavna poglavlja u kojima se proučavaju elementarni aspekti kompaktnosti. U prvom poglavlju definirali smo metriku i metričke prostore, proučavali nizove u metričkim prostorima, definirali pojmove kompaktan i potpun metrički prostor te karakterizirali kompaktnost preko potpunosti i potpune omeđenosti. U drugom poglavlju proučavali smo kompaktne topološke prostore. Definirali smo pojam otvorenog pokrivača te karakterizirali kompaktnost preko otvorenih pokrivača. U trećem poglavlju bavili smo se s kompaktnim skupovima u metričkim i topološkim prostorima. Povezali smo zatvorenost i kompaktnost te na kraju proučavali kompaktne skupove u \(\mathbb{R}^n\). The thesis is divided into three main chapters in which the elementary aspects of compactness are studied. In the first chapter, we defined metrics and metric spaces, studied sequences in metric spaces, defined the notions of a compact and a complete metirc space, and characterized compactness by completeness and complete boundedness. In the second chapter, we studied compact topological spaces. We defined the notions of an open cover and characterized compactness by open covers. In the third chapter, we dealt with compact sets in metric and topological spaces. We connected closedness and compactness and finally studied compact sets in \(\mathbb{R}^n\). |
