Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: As an introduction to point-free topology, we will explicitly show the connection between topology and frames (locales) and introduce an abstract notion, which in the point-free setting, can be thought of as a subspace of a topological space. In this setting, we refer to this notion as a sublocale and we will show that there are at least four ways to represent sublocales. By using the language of category theory, we proceed by investigating closure in the point-free setting by way of operators. We de ne what we mean by a coclosure operator in an abstract context and give two seemingly di erent examples of co-closure operators of Frm. These two examples are then proven to be the same. Compactness is one of the most important notions in classical topology and therefore one will nd a great number of results obtained on the subject. We will undertake a study into the interrelationship between three weaker compact notions, i.e. feeble compactness, pseudocompactness and countable compactness. This relationship has been established and is well understood in topology, but (to a degree) the same cannot be said for the point-free setting. We will give the frame interpretation of these weaker compact notions and establish a point-free connection. A potentially promising result will also be mentioned. / AFRIKAANSE OPSOMMING: As 'n inleiding tot punt-vrye topologie, sal ons eksplisiet die uiteensetting van hierdie benadering tot topologie weergee. Ons de nieer 'n abstrakte konsep wat, in die punt-vrye konteks, ooreenstem met 'n subruimte van 'n topologiese ruimte. Daar sal verder vier voorstellings van hierdie konsep gegee word. Afsluiting, deur middel van operatore, word in die puntvrye konteks ondersoek met behulp van kategorie teorie as taalmedium. Ons sal 'n spesi eke operator in 'n abstrakte konteks de nieer en twee o enskynlik verskillende voorbeelde van hierdie operator verskaf. Daar word dan bewys dat hierdie twee operatore dieselfde is. Kompaktheid is een van die mees belangrikste konsepte in klassieke topologie en as gevolg daarvan geniet dit groot belangstelling onder wiskundiges. 'n Studie in die verwantskap tussen drie swakker forme van kompaktheid word onderneem. Hierdie verwantskap is al in topologie bevestig en goed begryp onder wiskundiges. Dieselfde kan egter, tot 'n mate, nie van die puntvrye konteks ges^e word nie. Ons sal die puntvrye formulering van hierdie swakker konsepte van kompaktheid en hul verbintenis, weergee. 'n Resultaat wat moontlik belowend kan wees, sal ook genoem word.

Codensity monads are ubiquitous, as are various different notions of compactness and finiteness. Two such examples of "compact" spaces are compact Hausdorff Spaces and Linearly Compact Vector Spaces. Compact Hausdorff Spaces are the algebras of the codensity monad induced by the inclusion of finite sets in the category of sets. Similarly linearly compact vector spaces are the algebras of the codensity monad induced by the inclusion of finite dimensional vector spaces in the category of vector spaces. So in these two examples the notions of finiteness, compactness and codensity are intertwined. In this thesis we generalise these results. To do this we generalise the notion of ultrafilter, and follow the intuition of the compact Hausdorff case. We give definitions of general notions of "finiteness" and "compactness" and show that the algebras for the codensity monad induced by the "finite" objects are exactly the "compact" objects.

The measure of non-compactness is defined for any continuous mapping T : X Y between two Banach spaces X and Y as β(T) := inf { r > 0: T(BX) can be covered by finitely many open balls with radius r } . It can easily be shown that 0 ≤ β(T) ≤ ∥T∥ and that β(T) = 0, if and only if the mapping T is compact. My supervisor prof. Stanislav Hencl has proved in his paper that the measure of non-compactness of the known embedding W k,p 0 (Ω) → Lp∗ (Ω), where kp is smaller than the dimension, is equal to its norm. In this thesis we prove that the measure of non-compactness of the embedding between function spaces is under certain general assumptions equal to the norm of that embedding. We apply this theorem to the case of Lorentz spaces to obtain that the measure of non-compactness of the embedding Wk 0 Lp,q (Ω) → Lp∗,q (Ω) is for suitable p and q equal to its norm. 1

Made available in DSpace on 2015-05-15T11:46:20Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 700931 bytes, checksum: e90ff8d817f64f35c7f45fb88026619e (MD5) Previous issue date: 2014-04-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we will use the Mountain Pass Theorem, Ekeland s Variational Principle, the Concentration-Compactness Principle, the Brezis & Nirenberg Method, Penalization Method and some properties involving Nehari manifolds to obtain existence and multiplicity of solutions for the following class of elliptic systems. () 8<: u + V (x)u + u = r(x; u) em R3; = u2 em R3; where r : R3 R ! R is a function that has critical growth. / Neste trabalho, usaremos o Teorema do Passo da Montanha, Princípio Variacional de Ekeland, o Princípio de Concentração de Compacidade, o Método de Brezis & Nirenberga, o Método de Penalização e propriedades envolvendo Variedades de Nehari para obter resultados de existência e multiplicidade de soluções positivas para uma classe de sistemas elípticos ( também conhecidos como sistemas do tipo Schrödinger- Poisson)(-) 8<: -u + V (x)u + u = r(x; u) em R3; = u2 em R3; onde r : R3 R ! R é uma função que possui crescimento crítico.

Assumindo a existência de $\\mathfrak c$ ultrafiltros seletivos dois a dois incomparáveis (segundo a ordem de Rudin-Keisler) provamos que o grupo abeliano livre de cardinalidade $\\mathfrak c$ admite uma topologia de grupo enumeravelmente compacta com uma seqüência não trivial convergente. Sob as mesmas hipóteses, mostramos que um grupo topológico abeliano quase livre de torção $(G, +, \\tau)$ com $|G| = |\\tau| = \\mathfrak c$ admite uma topologia independente de $\\tau$ que o torna um grupo topológico e caracterizamos algebricamente os grupos abelianos de não torção que têm cardinalidade $\\mathfrak c$ e que admitem uma topologia de grupo enumeravelmente compacta (sem seqüências não triviais convergentes). Provamos, ainda, que o grupo abeliano livre de cardinalidade $\\mathfrak c$ admite uma topologia de grupo que torna seu quadrado enumeravelmente compacto e construímos um semigrupo de Wallace cujo quadrado é, também, enumeravelmente compacto. Por fim, assumindo a existência de $2^{\\mathfrak c}$ ultrafiltros seletivos, garantimos que se um grupo abeliano de não torção e cardinalidade $\\mathfrak c$ admite uma topologia de grupo enumeravelmente compacta, então o mesmo admite $2^{\\mathfrak c}$ topologias de grupo enumeravelmente compactas (duas a duas não homeomorfas). / Assuming the existence of $\\mathfrak c$ pairwise incomparable selective ultrafilters (according to the Rudin-Keisler ordering) we prove that the free abelian group of cardinality $\\mathfrak c$ admits a countably compact group topology that contains a non-trivial convergent sequence. Under the same hypothesis, we show that an abelian almost torsion-free topological group $(G, +, \\tau)$ with $|G| = |\\tau| = \\mathfrak c$ admits a group topology independent of $\\tau$ and we algebraically characterize the non-torsion abelian groups of cardinality $\\mathfrak c$ which admit a countably compact group topology (without non-trivial convergent sequences). We also prove that the free abelian group of cardinality $\\mathfrak c$ admits a group topology that makes its square countably compact and we construct a Wallace\'s semigroup whose square is countably compact. Finally, assuming the existence of $2^$ selective ultrafilters, we ensure that if a non-torsion abelian group of cardinality $\\mathfrak c$ admits a countably compact group topology, then it admits $2^$ (pairwise non-homeomorphic) countably compact group topologies.

O problema de Scarborough-Stone consiste em perguntar se o produto de espaços topológicos sequencialmente compactos precisa ser enumeravelmente compacto. Nesse trabalho estudamos alguns resultados que surgiram tentando resolver tal problema. Começamos com uma resposta negativa em ZFC usando espaços T2 e depois especificamos melhor condições sobre os axiomas de separação envolvendo os espaços do produto. Veremos respostas positivas envolvendo alguns axiomas de separação mais fortes como T6 (usando MA e a negação de CH) e T5 (usando o PFA). Além disso construímos mais respostas negativas usando construções como a Reta de Ostaszewski, espaços de Franklin-Rajagopalan e estruturas envolvendo álgebras Booleanas. / The Scarborough-Stone problem asks if every product of sequentially compact spaces must be a countably compact space. In this work we study some results that have arisen in attempt to solve this problem. We start our results with a negative answer in ZFC using T2 spaces and specify our conditions about the separability axioms of the spaces of the product. We will see positive answers assuming stronger separability axioms like T6 (using MA and the negation of CH) and T5 (using the PFA). We also construct more negative answers using constructions like the Ostaszewski line, Franklin-Rajagopalan spaces and structures involving Boolean algebras.

This thesis is devoted to show various applications of fixed point theorems on dif- ferential equations. In the beginning we use a notion of topological degree to derive several fixed points theorems, primarily Brouwer, Schauder and Kakutani-Ky Fan the- orem. Then we apply them on a wide range of relatively simple problems from ordinary and partial differential equations (ode and pde). Finally, we take a look on a few more complex problems. First is an existence of a solution to the model of mechanical os- cillator with non-monotone dependence of both displacement and velocity. Second is a solution to so called Gause predator-prey model with a refuge. The last one is cer- tain partial differential equation with a constraint which determines maximal monotone graph. 1

No description available.

Let G be a subset of the dual of a real Banach space X and F ⊂ G. Then F is a James boundary of G if each w∗ -continuous linear functional on X attains its supremum over G on an element of the set F. We ask whether a norm bounded subset of X which is countably compact for the topology generated by F is ne- cessary sequentially compact for the topology generated by G. The main content of our work is a positive solution to this problem. As a corollary we obtain James characterization of weakly compact subsets of a real Banach space. Due to the Eberlein-Šmuljan theorem a positive solution to the so called boundary problem is shown as a special case of the affirmative answer to the question raised above. The question is further discussed for a case of Banach spaces defined over the complex field. In this case we cannot use the old definition of the James boun- dary but by a "natural" way it is possible to redefine the term James boundary and then we are able to answer our question positively again. 1

In this thesis, we study the existence of random periodic solutions for both nonlinear dissipative stochastic functional differential equations (SFDEs) and semilinear nondissipative SFDEs in C([-r,0],R^d). Under some sufficient conditions for the existence of global semiflows for SFDEs, by using pullback-convergence technique to SFDE, we obtain a general theorem about the existence of random periodic solutions. By applying coupled forward-backward infinite horizon integral equations method, we perform the argument of the relative compactness of Wiener-Sobolev spaces in C([0,τ],C([-r,0]L²(Ω))) and the generalized Schauder's fixed point theorem to show the existence of random periodic solutions.