Title Razni aspekti povezanosti Title (english) Various aspects of connectivity Author Lara Milić Mentor Zvonko Iljazović (mentor)Committee member Zvonko Iljazović (predsjednik povjerenstva)Committee member Hrvoje Planinić (član povjerenstva)Committee member Ljiljana Arambašić (član povjerenstva)Committee member Snježana Lubura Strunjak (član povjerenstva)Granter University of Zagreb Defense date and country 2023-05-24, Croatia Scientific / art field, NATURAL SCIENCES Abstract Ovaj smo diplomski rad podijelili na dva veća poglavlja u kojima smo promatrali razne aspekte povezanosti. U prvom smo poglavlju ovog rada proučavali definicije norme i metrike te ekvivalentne metrike u kontekstu metričkih prostora. Također smo istraživali otvorene i zatvorene skupove u tim prostorima, kao i neprekidne funkcije između njih. Nakon toga smo uveli pojmove topologije i topološkog prostora, te smo detaljnije proučavali baze topologije i neprekidne funkcije izmedu topoloških prostora. Nadalje, obradivali smo potprostore metričkih i topolo ških prostora. U drugom poglavlju smo definirali pojam separacije u topološkim i metričkim prostorima, kao i povezane topološke i metričke prostore. Dokazali smo da je svaki segment [a,b] povezan, a zatim smo proučavali komponente povezanosti točke u topološkom prostoru i neka njihova svojstva. Također smo definirali produkt topoloških prostora i dokazali da je produkt dva povezana topološka prostora povezan. Uveli smo pojam putevima povezanog topološkog prostora, dokazali da je svaki putevima povezan topološki prostor povezan i dali primjer topološkog prostora koji je povezan, ali nije putevima povezan. Nadalje, istaživali smo lokalnu povezanost i naveli primjer topološkog prostora koji je povezan, ali nije lokalno povezan. Na kraju smo dokazali Baireov teorem o kategoriji i pomoću njega teorem kojim se tvrdi da se povezan, lokalno povezan i potpun metrički prostor ne može prikazati kao unija prebrojivo mnogo zatvorenih, pravih i medusobno disjunktnih podskupova.
Abstract (english) We divided this thesis into two larger chapters in which we observed various aspects of connection. In the first chapter of this work, we studied the definitions of norm, metric and equivalent metric in the context of metric spaces. We also explored open and closed sets in these spaces, as well as continuity of functions between them. After that, we introduced the term of topology and topological spaces. Also, we studied the bases of a topology and continuity of functions between topological spaces. Furthermore, we researched subspaces of metric and topological spaces. In the second chapter, we defined the notion of separation in topological and metric spaces, as well as connected topological and metric spaces. We proved that every segment [a,b] is connected. Then we studied the connected components of a node in topological spaces and some of their properties. We also defined the product of topological spaces and proved that the product of two connected topological spaces is connected. We introduced the notion of a path-connected topological space, proved that every path-connected topological space is connected topological space, and gave an example of a topological space that is connected but not path-connected. Furthermore, we investigated local connectivity and provided an example of a topological space that is connected but not locally connected. Finally, we proved Baire’s category theorem and using it the theorem that claims that a connected, locally connected, and complete metric space cannot be represented as the union of countably many closed, real, and mutually disjoint subsets.
Keywords
definicije norme i metrike
ekvivalenti metrike
kontekst metričkih prostora
topologija
topološki prostor
Baireov teorem o kategoriji
Keywords (english)
definitions of norm
metric
equivalent metric
metric spaces
topology
topological spaces
Baire’s category theorem
Language croatian URN:NBN urn:nbn:hr:217:220156 Study programme Title: Mathematical Statistics Type of resource Text File origin Born digital Access conditions Open access Terms of use Created on 2024-02-09 13:11:12
