Ekstremno nepovezani prostori

delo diplomskega seminarja

Maruša Lekše (Author), Aleš Vavpetič (Mentor)

Abstract

Ekstremno nepovezani prostori so prostori, v katerih je zaprtje vsake odprte množice odprta množica. V tem diplomskem delu se osredotočimo na nekatere zanimive lastnosti ekstremno nepovezanih prostorov Tihonova. Pokažemo, da je v takšnih prostorih vsako konvergentno zaporedje od nekega člena naprej konstantno. Doka- žemo, da je prostor Tihonova ekstremno nepovezan natanko tedaj, ko ima vsaka navzgor omejena podmnožica delno urejene množice C(X) natančno zgornjo mejo. Pokažemo tudi, da je za prostore Tihonova vsako ekstremno nepovezano topološko polje diskretno in da je topološka grupa diskretna natanko tedaj, ko je prostor G×G ekstremno nepovezan. Na koncu s pomočjo Stone-Čechove kompaktifikacije najdemo primer prostorov, ki jih preučujemo.

Keywords

ekstremno nepovezan prostor;topološke grupe;topološka polja;Stone-Čechova kompaktifikacija;

Data

Language:	Slovenian
Year of publishing:	2020
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[M. Lekše]
UDC:	515.1
COBISS:	58825987
Views:	1033
Downloads:	117
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Extremally disconnected spaces
Secondary abstract:	Extremally disconnected spaces are spaces, in which the closure of every open set is open. In this thesis we examine some interesting properties of extremally disconnec- ted Tychonoff spaces. We prove that in such spaces, every convergent sequence is constant from some point on. We show that a Tychonoff space is extremally disco- nected if and only if every subset of the partially ordered set C(X) that has an upper bound, also has a least upper bound. We also prove that an extremally disconnected Tychonoff topological field is discrete and that an extremally disconnected Tycho- noff topological group G is discrete if and only if the product G × G is extremally disconnected. In the end of this thesis we define Stone-Cech compactification in order to find an example of the spaces that we are studying.
Secondary keywords:	extremally disconnected space;topological groups;topological fields;Stone-Čech compactification;
Type (COBISS):	Final seminar paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	29 str.
ID:	12008211