Tihonovljev teorem

master's thesis

• Title (english): Tychonoff theorem
• Author: Tomislav Crnogorac
• Mentor: Maja Resman (mentor)
• Committee member: Maja Resman (predsjednik povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Aleksandra Čižmešija (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-03-03, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Glavni cilj ovog rada je iskazati i predstaviti neke od dokaza Tihonovljevog teorema, koji tvrdi da je proizvoljni produkt kompaktnih topoloških prostora kompaktan u produktnoj topologiji. Veći dio rada, prva tri poglavlja, pripremaju teren upravo za to. Prvo poglavlje služi kao svojevrsni rječnik pojmova koji se tiču strukture topološkog prostora i direktnog produkta topoloških prostora. U drugom poglavlju kratko se promatraju nizovi u općenitom topološkom prostoru, a zatim se uvode i istražuju poopćenja nizova, a to su mreže i filtri. Oni, za razliku od nizova, omogućuju dovoljnu općenitost kada je u pitanju konvergencija u topološkom prostoru. Treće poglavlje sadrži osnovne činjenice vezane uz kompaktnost topološkog prostora i neke primjere takvih prostora. Osim toga, iznosimo i karakterizacije kompaktnosti preko mreža i filtara iz prethodnog poglavlja. Konačno, u zadnjem poglavlju iskazuje se Tihonovljev teorem i dokazuje se na dva načina: preko mreža i filtara, odnosno preko centriranih familija (familija sa svojstvom konačnog presjeka).

Abstract (english)

The main goal of this paper is to state and present some of the proofs of the Tychonoff theorem, which states that an arbitrary product of compact topological spaces is compact in the product topology. The first three chapters of this thesis prepare the setting just for its proofs. The first chapter serves as a kind of a dictionary of terms concerning the structure of topological spaces and the direct product of topological spaces. In the second chapter, sequences in general topological space are briefly discussed and then generalized introducing the notions of nets and filters. They, unlike sequences, allow sufficient generality when it comes to convergence in a topological space. The third chapter contains basic facts about the compactness of a topological space and examples of such spaces. In addition, it contains characterization of compactness using nets and filters from the previous chapter. Finally, in the last chapter, Tychonoff theorem is stated and two proofs are presented: the first one using nets and filters, and the second one using collections the with finite intersection property.

• Keywords: topološki prostor; mreže; filtri; kompaktnost topološkog prostora; familija sa svojstvom konačnog presjeka
• Keywords (english): topological spaces; nets; filters; compactness of a topological space; collections with finite intersection property
• Language: croatian
• URN:NBN: urn:nbn:hr:217:996675
• 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: 2022-03-25 13:23:54