Hahn-Mazurkiewiczev teorem

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.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