Dugundjijev razširitveni izrek : delo diplomskega seminarja

Urh, Gašper (author), Smrekar, Jaka (mentor)

V diplomskem delu je podrobno obravnavan dokaz Dugundjijeve razširitve Tietzejevega razširitvenega izreka za preslikave iz zaprte podmnožice metričnega prostora v lokalno konveksne topološke vektorske prostore in njegove neposredne posledice v normiranih topoloških vektorskih prostorih. Opisan je tudi dokaz za primer, ko namesto metričnega prostora opazujemo parakompakten Hausdorffov prostor. Definirana je Dugundjijeva razširitvena lastnost, ki zajema obstoj simultanega razširitvenega operatorja, in dokazano je, da imajo poleg metričnih to lastnost tudi stratifikabilni prostori. S primerom Sorgenfreyeve premice je prikazano, da tak simultani razširitveni operator lahko najdemo tudi v nekaterih nestratifikabilnih prostorih, s primerom Michaelove premice pa, da te lastnosti nimajo vsi normalni prostori, tudi če so parakompaktni in Hausdorffovi.

• Language: Slovenian
• Keywords: razširitveni izreki, simultani razširitveni operator, parakompaktnost, stratifikabilni prostori, posplošeni linearno urejeni prostori
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
• PID: 20.500.12556/RUL-118217
• UDC: 515.1
• COBISS.SI-ID: 58828035
• Publication date in RUL: 27.08.2020

URH, Gašper, 2020, Dugundjijev razširitveni izrek : delo diplomskega seminarja [online]. Bachelor’s thesis. [Accessed 6 March 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=118217

Secondary language

• Language: English
• Title: Dugundji extension theorem

Abstract

In this bachelor's thesis, the proof of Dugundji’s extension of Tietze extension theorem for mappings from a closed subset of a metric space into a locally convex linear space is presented in detail together with its immediate consequences. The theorem where a paracompact Hausdorff space is considered instead of a metric one is also proved. The Dugundji extension property concerning the existence of a simultaneous extender is defined and it is proved that not only metric but also stratifiable spaces have this property. With the example of the Sorgenfrey line it is shown that it is sometimes possible to find such simultaneous extenders in spaces that are not stratifiable, while with the example of the Michael line it is shown that not all normal spaces have this property, even if they are paracompact Hausdorff spaces.

• Keywords: extension theorems, simultaneous extender, paracompactness, stratifiable spaces, generalized ordered spaces