
Brouwer fixed point theorem (proofs and history)
Vítek, Tomáš ; Hušek, Miroslav (advisor) ; Vejnar, Benjamin (referee)
The aim of this work was to present different approaches to the proof of Brouwer fixed point theorem and to avoid proofs based on homotopy theory, degree of mapping or any nontrivial algebraic topology. The proofs were chosen so that only a basic knowledge of combinatorics and mathematical analysis is required to understand them and the reader could learn about other fundamental topological theorems. At first, we prove by a combinatorial procedure BorsukUlam theorem from which Brouwer theorem simply follows. We then use the basics of mathematical analysis to prove a theorem known as The hairy ball problem, which also directly implies Brouwer theorem. Finally, we will show an unconventional application of Brouwer theorem to prove the fundamental theorem of algebra. 1


Hcompactifications of topological spaces
Tížková, Tereza ; Vejnar, Benjamin (advisor) ; Hušek, Miroslav (referee)
Hcompactifications form an important type of compactifications, carrying the ex tra property that all automorphisms of a given topological space can be continuously extended over such compactifications. Van Douwen proved there are only three Hcompactifications of the real line and only one of the rationals. Vejnar proved that there are precisely two Hcompactifications of higher dimensional Euclidean spaces. The concept of Hcompactifications is introduced at the beginning, extra emphasis being put on the Alexandroff and StoneČech compacti fication. We summarize findings that exist about Hcompactifications of some wellknown spaces. The result we come with in the Chapter 3 is that there is only one Hcompactification of the set of all rational sequences, which is precisely the StoneČech compactification. The third chapter describes various properties of the set of all rational sequences and its clopen subsets. Some of them  mainly strong zerodimensionality and strong homogeneity  are then used to reach the said result. In the final Chapter 4, we ask a question about the set of all Hcompactifications of the Hilbert space of all square summable real sequences and propose three ways to tackle this problem. We show that under certain conditions, any Hcompactification of a space is homeomorphic to...


Extensions of functions from subspaces of metric spaces
Hevessy, Michal ; Hušek, Miroslav (advisor) ; Vejnar, Benjamin (referee)
Function extension is a classical problem in mathematics. In this thesis we look into an extesion of realvalued functions defined on metric spaces. The first chapter is intro ductory and describes extension problem. In the second one we discuss a known method for extension of special family of uniformly continuous functions and show that the me thod can be modified for continuous functions. The third chapter examines a method for extension of continuous functions described by Whitney. Finally, in the last chapter we show a characterisation of uniformly continuous function, having uniformly continuous extensions. 1


Hcompactifications of topological spaces
Tížková, Tereza ; Vejnar, Benjamin (advisor) ; Hušek, Miroslav (referee)
Hcompactifications form an important type of compactifications, carrying the ex tra property that all automorphisms of a given topological space can be continuously extended over such compactifications. Van Douwen proved there are only three Hcompactifications of the real line and only one of the rationals. Vejnar proved that there are precisely two Hcompactifications of higher dimensional Euclidean spaces. The result we come with in the Chapter 2 is that there is only one Hcompactification of the set of all rational sequences, which is precisely the StoneČech compactification. For the proof, we use strong zerodimensionality, strong homogeneity and other properties of the set of all rational sequences and its clopen subsets. In the Chapter 3, we ask an ambitious question about the set of all Hcompactifications of the Hilbert space of all square summable real sequences and propose some ways to tackle this problem, e.g. characterizations of the StoneČech compactification or tools used to describe Hcompactifications of the real space of dimension 2. In the final chapter, we analyze the set of all Hcompactifications of a space using a categorytheoretic approach and study properties of categories of Hcompactifications and functors in such categories. 1


Homogeneity of topological structures
Vejnar, Benjamin ; Hušek, Miroslav (advisor) ; Pyrih, Pavel (referee)
In the present work we study those compacti cations such that every autohomeomorphism of the base space can be continuously extended over the compacti cation. These are called Hcompacti cations. We characterize them by several equivalent conditions and we prove that Hcompacti cations of a given space form a complete upper semilattice which is a complete lattice when the given space is supposed to be locally compact. Next, we describe all Hcompacti cations of discrete spaces as well as of countable locally compact spaces. It is shown that the only Hcompacti cations of Euclidean spaces of dimension at least two are onepoint compacti cation and the CechStone compacti cation. Further we get that there are exactly 11 Hcompacti cations of a countable sum of Euclidean spaces of dimension at least two and that there are exactly 26 Hcompacti cations of a countable sum of real lines. These are all described and a Hasse diagram of a lattice they form is given.

 
 

Variations of Banach fix point theorem
Pošta, Petr ; Hušek, Miroslav (advisor) ; Lukeš, Jaroslav (referee)
\azev prace: Yariaee Banachovy vety o pevnem bode Autor: Potr Posta Katecha (ustav): Katedra malematieke analy/y Vedouci bakalarske pn'uo: prof. R.NDr. Miroslav Husek. DrSr. email vedouciho: nihnsek'fika.rlin.mff.cuni.c/ Abstrakt: V predlozene pra.ci studujcmo rozlicno dusledky a /ohccnfjiii Bana chovy vrty o pcvnrni hodr. V prvni Oasli sliulujciin' diislcdky klasickrlio Bana cliDva prhiripu kuiitrakcc: posloiipnosti kunlraktivnicli zo)j]'ax,(ini, ru/.iie variact1 podnn'iiky koiit.rakt.iviiost.i xobra/cni. pffkladv pou/.iti v Ranacliovych prostorodi. diskrrl.ni prinrip koiilrakcc (Filriilxn'^uva a Jachyinskrho veiv.r) a tit.a/ku ckviva. Icncc diskrutniYh vet .s Baiiachovou \vtou. V druhr casli jsou nastinriiy moxnr prfstupy k zobrcuc'iii liaiiachovy vely: jako ph'klady jsuu dokazany ruzne vrty o pevuriu liodr (autory jsou Edrlstcin, Bailey. Civir, Kirk a dalsf), ktr.n'1 xoheciiuji Banachovii vOlu. Kh'cova sluva: Bauacliova vela u kunt.ra.kci. konl.iakcc, prvny bod, /obc'dinnr kon Title: Variations of Bauarh iix point tluMirrin Author: Potr I'ost.a Do]>artim'iit.: Dopart.mont of iMa.lhonia.tica.l Analysis Suporvisor: prof. RNDr. Miroslav Ilvisck. DrSc. Su]>ervisor's cniail addrcsw: Abstract: In the prosrnt \\ork wo study various consequences and generalizations of Bana.ch tixcd point tlieor(nii. In...


Extension of mappings into Banach spaces
Novotný, Vojtěch ; Hušek, Miroslav (advisor) ; Pyrih, Pavel (referee)
This diploma thesis deals with extending continuous and uniformly continuous mappings. It studies Lebesgue's and Tietze's work in metric spaces through Urysohn's theorem in normal topological spaces, Kat etovs' papers about uniformly continuous functions up to Dugundji's theorem and relationship between continuous extending of pseudometrics and mappings. It connects the articles of nineteen mathematicians of the twentieth century, presents plenty of theorems in more general form and shows that they could be formulated earlier or proved in another way.


Metric spaces of Hölder mappings
Novotný, Vojtěch ; Hušek, Miroslav (advisor) ; Spurný, Jiří (referee)
Nazcv prace: Metricke prostory hoklerovskych funkci Autor: Vojtech .\ovotny Katcdra: Kiitcrlra matenmticke analyzy Vedouci bakalafske pracc: Prof. liNDr. Miruslav lln.sek, DrSc. Eniail vedouciho: mhusek@karlin.mff.cuni.cz Abstrakt: Ukolem bakalafske prace je definovat prostory aholderovskych zobrazeni s pfirozenou pseudonormou a studovat jejich vlastuosti. Nesepa rahilitu, uplnost ci v/tahy jednotlivyoh prostoru vysetfujeme nejdfivc na realnych holdorovskycli fuukcich defiuovaiiych na [0, 1], po/xlcji na jinych omezeuych u/.avfeiiych, ...'li i otevfenych intervalech a nakonec na cole reahu'1 pfiince. V /avererne ka.pitole ziskanc vyslcdky zobecnujeine ]>ro ho'ldcrovska /obrazcni v obecnejsich nietriekyrh prostorech. Klicova slova: holdcrovske zobrazeni, ... zobrazoni. niotricky prostor Title: Metric spaces of Holder mappings Author: Vojtech .\ovotny Department: Department of Mathematical Analysis Supervisor: Prof. KA'Dr. Miroslav Ilusek, DrSc. Supervisor's email address: mhusek@karlin.mff.cuni.cz Abstract: The task of the bachelor work is to define spaces of oHolder map pings with their natural pscudononn and to study their properties. First we investigate nonseparability. completeness and relations between particular spaces of real Holder functions defined on [0. l], later...
