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master's thesis
Baireovi prostori i primjene
Zagreb: University of Zagreb, Faculty of Science, 2019. urn:nbn:hr:217:150666
Ivičinec, Matija
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Ivičinec, M. (2019). Baireovi prostori i primjene (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:150666

Ivičinec, Matija. "Baireovi prostori i primjene." Master's thesis, University of Zagreb, Faculty of Science, 2019. https://urn.nsk.hr/urn:nbn:hr:217:150666

Ivičinec, Matija. "Baireovi prostori i primjene." Master's thesis, University of Zagreb, Faculty of Science, 2019. https://urn.nsk.hr/urn:nbn:hr:217:150666

Ivičinec, M. (2019). 'Baireovi prostori i primjene', Master's thesis, University of Zagreb, Faculty of Science, accessed 23 June 2024, https://urn.nsk.hr/urn:nbn:hr:217:150666

Ivičinec M. Baireovi prostori i primjene [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2019 [cited 2024 June 23] Available at: https://urn.nsk.hr/urn:nbn:hr:217:150666

M. Ivičinec, "Baireovi prostori i primjene", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2019. Available at: https://urn.nsk.hr/urn:nbn:hr:217:150666

Metadata
TitleBaireovi prostori i primjene
AuthorMatija Ivičinec
MentorIlja Gogić (mentor)
Committee memberIlja Gogić (predsjednik povjerenstva)
Committee memberRudi Mrazović (član povjerenstva)
Committee memberVjeran Hari (član povjerenstva)
Committee memberEduard Marušić-Paloka (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2019-07-15, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
Baireov teorem o kategoriji važan je teorem u matematici koji se često primjenjuje pri dokazivanju egzistencije elemenata koje je teško vizualizirati. Grubo govoreći, skupove metričkog prostora svrstavamo u tri kategorije. Skupovi prve kategorije su skupovi koje smatramo "malima" odnosno zanemarivima u promatranom prostoru. Skupovi druge kategorije su oni skupovi koji nisu "mali", dok su rezidualni skupovi oni koje smatramo "velikima". Baireovi prostori definirani su kao metrički prostori u kojima je svaki rezidualan skup gust. Baireov teorem o kategoriji kaže da je svaki potpun metrički prostor Baireov prostor. U ovom radu dokazujemo Baireov teorem o kategoriji te dajemo razne primjere vezane uz spomenute koncepte. U zadnjem poglavlju rada prikazane su neke zanimljive primjene Baireovog teorema o kategoriji. Između ostalog, dokazujemo da su generičke neprekidne funkcije \(f:[a,b]\rightarrow \mathbb{R}\) nigdje derivabilne.
Abstract (english)
The Baire category theorem is an important theorem in mathematics which is often used in proving existence of elements which are hard to visualize. Speaking loosely, sets of a metric space are divided into three categories. Sets of the first category are sets that are considered as ”small” or negligible in the observed space. Sets of the second category are those that are not ”small”, while the residual sets are considered ”large”. We define a Baire space as a metric space in which every residual set is dense. The Baire category theorem states that every complete metric space is a Baire space. In this thesis we prove the Baire category theorem and we present various examples of the mentioned concepts. In the last chapter of the thesis we present some interesting applications of the Baire category theorem. Among other things, we prove that generic continuous functions \(f:[a,b]\rightarrow \mathbb{R}\) are nowhere differentiable.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:150666
Study programmeTitle: Mathematics Education; specializations in: Mathematics Education
Course: Mathematics Education
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2019-09-30 08:55:56