Hahn-Mazurkiewiczev teorem

Vukić, Slavica (2015) Hahn-Mazurkiewiczev teorem. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

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.

Item Type: Thesis (Diploma thesis)
Supervisor: Iljazović, Zvonko
Date: 2015
Number of Pages: 35
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 29 Oct 2015 12:26
Last Modified: 29 Oct 2015 12:26
URI: http://digre.pmf.unizg.hr/id/eprint/4198

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