Baireovi prostori i primjene

master's thesis

• Author: Matija Ivičinec
• Mentor: Ilja Gogić (mentor)
• Committee member: Ilja Gogić (predsjednik povjerenstva)
• Committee member: Rudi Mrazović (član povjerenstva)
• Committee member: Vjeran Hari (član povjerenstva)
• Committee member: Eduard Marušić-Paloka (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-07-15, 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: Baireov teorem; skupovi metričkog prostora; Baireov teorem o kategoriji
• Keywords (english): Baire category theorem; sets of a metric space
• Language: croatian
• URN:NBN: urn:nbn:hr:217:150666
• Study programme: Title: 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 resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-09-30 08:55:56