Konvergencije nizova skupova u metričkim i topološkim prostorima

master's thesis

• Title (english): Convergences of sequences of sets in metric and topological spaces
• Author: Filip Žulec
• Mentor: Zvonko Iljazović (mentor)
• Committee member: Zvonko Iljazović (predsjednik povjerenstva)
• Committee member: Matija Kazalicki (član povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Committee member: Marko Radulović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-09-22, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu promatrali smo nizove skupova u metričkim i topološkim prostorima, te smo dali nekoliko definicija konvergencije takvih nizova. Prvo smo iskazali sve osnovne pojmove vezane uz metričke i topološke prostore koji su nam bili potrebni za daljnje razmatranje. Na početku smo limes niza podskupova (An)n∈N skupa X definirali kao skup A svih točaka prikazivih kao limesi nizova čiji n-ti član jest element skupa An. Taj limes nazvali smo limes prve vrste. Tu definiciju poopćili smo i na općeniti topološki prostor, a potom smo i niz skupova u toj definiciji poopćili na hiperniz. Jedno od svojstava limesa prve vrste omogućilo nam je da damo još jednu definiciju, po kojoj za svaku točku x ∈ X mora vrijediti d(x, An) → d(x, A). Taj limes nazvali smo limes druge vrste. U sljedećem koraku, promatrali smo Hausdorffovu metriku dH, te smo definirali limes niza skupova (An)n∈N s obzirom na tu metriku kao neprazan zatvoren skup A za koji vrijedi dH(An, A) → 0. Promatrali smo i konvergenciju s obzirom na Vietorisovu topologiju, te smo dali i definiciju analognu definciji konvergencije niza točaka u topološkom prostoru. Na kraju, promatrali smo funkcije na nizovima skupova. Pokazali smo da, ako je funkcija neprekidna na svakom od skupova u nizu, ne mora nužno biti neprekidna na njegovom limesu prve vrste, pa čak ni na interioru tog skupa. Ipak, dali smo još jednu definciju konvergencije niza skupova, pomoću konvergencije niza karakterističnih funkcija tih skupova po točkama.

Abstract (english)

In this work, we observed sequences of sets in metric and topological spaces, and we gave several defnitions of convergence of such sequences. First, we stated all the basic terms involving metric and topological spaces which were necessary for further observation. Initially, we defined the limit of a sequence (An)n∈N of subsets of X as the set A of all points that are limits of a sequence (xn)n∈N whose n-th term is an element of An. We called that limit the limit of the first kind. We generalized this definiton onto topological spaces, and then we generalized the sequences in that definition onto nets. One of the properties of the limit of the first kind helped us to make another definition, by which each point x ∈ X must satisfy d(x, An) → d(x, A). We called that limit the limit of the second kind. In the next step, the observed the Hausdorff metric dH, and we defined the limit of a sequence (An)n∈N of sets considering said distance as the non-empty closed set A which satisfies dH(An, A) → 0. We also observed convergence considering the Vietoris topology, and we gave a definition analogous to the definition of a limit of a sequence in a topological space. Lastly, we observed functions on sequences of sets. We proved that, if a function is continuous on each set in a sequence, it is not necessarily continuous on its limit of the first kind, or even in the interior of that set. Still, we gave one more definition of convergence of a sequence of sets, using pointwise convergence of characteristic functions of the given sets.

• Keywords: limes prve vrste; Hausdorffova metrika; Vietorisova topologija
• Keywords (english): limit of the first kind; Hausdorff metric; Vietoris topology
• Language: croatian
• URN:NBN: urn:nbn:hr:217:728290
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-02-07 12:51:53