Hahn-Mazurkiewiczev teorem

master's thesis

• Author: Slavica Vukić
• Mentor: Zvonko Iljazović (mentor)
• Committee member: Zvonko Iljazović (predsjednik povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Committee member: Dragutin Svrtan (član povjerenstva)
• Committee member: Mladen Vuković (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-07-13, 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: Hahn-Mazuriewiczev teorem; Hausdorffov prostor; Peanov kontinuum; Peanova krivulja; Urysonov teorem metrizacije; Aleksandrov-Hausdorffov teorem; Cantorov skup
• Keywords (english): Hahn-Mazurkiewicz theorem; Hausdorff space; Peano continuum; Peano curve; Urysohn metrization; Aleksandroff-Hausdorff theorem; Cantor set
• Language: croatian
• URN:NBN: urn:nbn:hr:217:491191
• 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: 2019-02-20 10:59:47