Hilbertova kocka

master's thesis

• Author: Kristina Buljević
• Mentor: Zvonko Iljazović (mentor)
• Committee member: Zvonko Iljazović (predsjednik povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Committee member: Nikola Sandrić (član povjerenstva)
• Committee member: Zvonimir Bujanović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj diplomski rad je osmišljen na način da nakon što uvedemo sve pojmove i "pripremimo" teren za proučavanje većih svojstava zapravo dobijemo važne rezultate. Prvo poglavlje se bavi općenito pomovima iz topologije poput otvorenih skupova, zatvaraču, kompaktnosti, neprekidnosti, filterima... U drugom poglavlju na parcijalno uređenom skupu, uz pomoć svojstava koje ima, proučava se kada neki skup ima maksimalan element te se dolazi do Zornove leme i njenih posljedica. Treće poglavlje opisuje stvaranje topologije na produktu, uvode se projekcije, ispituju svojstva produktne topologije(kompaktnost, separabilnost, metrizabilnost) i dolazi do zaključka da je važan i indeksni skup za konstrukciju i svojstva samog produkta. Iako smo krenuli od pojedinog topološkog prostora, na kraju ipak zaključujemo da je Hilbertova kocka osim produkta zapravo i skup svih funkcija sa skupa prirodnih brojeva na interval od 0 do 1.

Abstract (english)

This graduate thesis is designed in such a way that after we introduce all the terms and ”prepare” the ground for studying the larger properties we actually get important results. The first chapter deals generally with topology exits such as open sets, shutter, compactness, continuity, filters ... In the second chapter on a partially arranged set, with the help of its properties, it is studied when some set has the maximum element and comes to Zorn’s lemma and its consequences. Chapter 3 describes the creation of topology on the product, introduces projections, examines the properties of the product topology (compactness, separability, metrizability) and concludes that an index and a set of properties for the construction and properties of the product are also important. Although we have gone from a particular topological space, we conclude, in the end, that Hilbert’s cube, apart from the product, is actually a set of functions from a set of natural numbers at an interval of 0 to 1.

• Keywords: topološki prostori; Zornova lema; Kartezijev produkt topoloških prostora
• Keywords (english): topological spaces; Zorn's lemma; Cartesian product of topological spaces
• Language: croatian
• URN:NBN: urn:nbn:hr:217:932341
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2017-12-20 12:49:54