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master's thesis
Hahn-Mazurkiewiczev teorem
Zagreb: University of Zagreb, Faculty of Science, 2015. urn:nbn:hr:217:491191
Vukić, Slavica
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Vukić, S. (2015). Hahn-Mazurkiewiczev teorem (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:491191

Vukić, Slavica. "Hahn-Mazurkiewiczev teorem." Master's thesis, University of Zagreb, Faculty of Science, 2015. https://urn.nsk.hr/urn:nbn:hr:217:491191

Vukić, Slavica. "Hahn-Mazurkiewiczev teorem." Master's thesis, University of Zagreb, Faculty of Science, 2015. https://urn.nsk.hr/urn:nbn:hr:217:491191

Vukić, S. (2015). 'Hahn-Mazurkiewiczev teorem', Master's thesis, University of Zagreb, Faculty of Science, accessed 23 December 2024, https://urn.nsk.hr/urn:nbn:hr:217:491191

Vukić S. Hahn-Mazurkiewiczev teorem [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2015 [cited 2024 December 23] Available at: https://urn.nsk.hr/urn:nbn:hr:217:491191

S. Vukić, "Hahn-Mazurkiewiczev teorem", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2015. Available at: https://urn.nsk.hr/urn:nbn:hr:217:491191

Metadata
TitleHahn-Mazurkiewiczev teorem
AuthorSlavica Vukić
MentorZvonko Iljazović (mentor)
Committee memberZvonko Iljazović (predsjednik povjerenstva)
Committee memberPavle Pandžić (član povjerenstva)
Committee memberDragutin Svrtan (član povjerenstva)
Committee memberMladen Vuković (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2015-07-13, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
Hahn-Mazuriewiczev teorem jedan je od najvažnijih rezultata u povijesti opće topologije, jer u potpunosti rješava problem Peanovih krivulja. Teorem kaže da je Hausdorffov prostor neprekidna slika segmenta ako i samo ako je Peanov kontinuum, odnosno kompaktan, povezan i lokalno povezan metrički prostor. U ovom radu dajemo kratak opis spomenutog problema i predstavljamo jedan dokaz Hahn-Mazurkiewiczevog teorema. Činjenica da je neprekidna slika segmenta Peanov kontinuum jednostavna je posljedica Urysonovog teorema metrizacije i činjenice da je lokalna povezanost očuvana neprekidnim zatvorenim preslikavanjem. Za dokaz drugog smjera Hahn-Mazurkiewiczevog teorema iskoristit ćemo Aleksandrov-Hausdorffov teorem o postojanju neprekidne surjekcije Cantorovog skupa na proizvoljan kompaktan metrički prostor, te ćemo jedno takvo preslikavanje neprekidno proširiti na čitav segment.
Abstract (english)
The Hahn-Mazurkiewicz theorem is one of the most important results in the history of point-set topology, because it completely solves the problem of “space-filling” curves. It states that a Hausdorff space is a continuous image of a line segment if and only if it is a Peano continuum, that is, compact, connected and locally connected metric space. In this B.sc. thesis we give a short description of the problem and present a proof of this theorem. The fact that a continuous image of a line segment is a Peano continuum is a simple consequence of the Urysohn metrization theorem combined with the fact that local connectedness is preserved by a continuous closed map. To prove the other direction of our theorem we use the Aleksandroff-Hausdorff theorem about the existence of continuous mapping of the Cantor set onto an arbitrary compact metric space and we continuously extend this mapping over the line segment.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:491191
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 on2019-02-20 10:59:47