Topološki aspekti aksioma potpunosti

master's thesis

• Author: Martina Volarić
• Mentor: Zvonko Iljazović (mentor)
• Committee member: Zvonko Iljazović (predsjednik povjerenstva)
• Committee member: Sonja Štimac (član povjerenstva)
• Committee member: Maja Resman (član povjerenstva)
• Committee member: Zrinka Franušić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-07-17, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu proučavamo neprekidne funkcije na segmentu, ali i topološke prostore i koncepte s tim u vezi pomoću kojih možemo dokazati neke klasične rezultate vezane za neprekidne funkcije na segmentu. U prvom poglavlju koristeći gomilište nizova dokazujemo da svaka neprekidna funkcija na segmentu poprima minimum i maksimum. U drugom poglavlju proučavamo pojmove topologije, baze topologije, neprekidnosti funkcije između topoloških prostora i relativne topologije. Također, proučavamo kompaktne skupove u topološkim prostorima te dokazujemo da neprekidne realne funkcije na kompaktnim prostorima poprimaju minimum i maksimum. U trećem poglavlju proučavamo povezanost u topološkim prostorima te dokazujemo da neprekidna realna funkcija na povezanom prostoru poprima međuvrijednosti.

Abstract (english)

In this thesis we have studied continuous functions on segment, topological spaces and related concepts which we use to prove some classical results related to continuous functions on a segment. In the first chapter we use that each continuous function on a segment attains its minimum and maximum. In the second chapter we study topological spaces, continuous functions between topological spaces and relative topologies. We also study compact sets in topological spaces and we prove that each continuous real function on a compact space attains its minimum and maximum. In the third chapter we examine connectedness in topological spaces and we prove that each continuous real function on a connected space attains intermediate values.

• Keywords: neprekidne funkcije na segmentu; topološki prostori; gomilište nizova; kompaktni skupovi u topološkim prostorima
• Keywords (english): continuous functions on segment; topological spaces; accumulation point of a sequence; compact sets in topological spaces
• Language: croatian
• URN:NBN: urn:nbn:hr:217:597129
• 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: 2019-07-05 11:08:01