We begin by recalling the notion of a coarse space as defined by John Roe. We show that metrizability of coarse spaces is a coarse invariant. The concepts of bounded geometry, asymptotic dimension, and Guoliang Yu's Property A are investigated in the setting of coarse spaces. In particular, we show that bounded geometry is a coarse invariant, and we give a proof that finite asymptotic dimension implies Property A in this general setting. The notion of a metric approximation is introduced, and a characterization theorem is proved regarding bounded geometry. Chapter 7 presents a discussion of coarse structures on the minimal uncountable ordinal. We show that it is a nonmetrizable coarse space not of bounded geometry. Moreover, we show that this space has asymptotic dimension 0; hence, it has Property A.Finally, Chapter 8 regards coarse structures on products of coarse spaces. All of the previous concepts above are considered with regard to 3 different coarse structures analogous to the 3 different topologies on products in topology. In particular, we see that an arbitrary product of spaces with any of the 3 coarse structures with asymptotic dimension 0 has asymptotic dimension 0.

O conceito de reflexão em topologia expressa o fato de que um espaço satisfaz uma dada propriedade sempre que esta é satisfeita por seus subespaços \"menores\". Neste trabalho, estuda-se a reflexão de propriedades envolvendo a maioria das principais funções cardinais e metrizabilidade, bem como outras propriedades relacionadas. São discutidos problemas em aberto -- como o problema de Hamburger --, incluindo respostas parciais e exemplos de consistência. Várias dentre as demonstrações apresentadas utilizam técnicas de submodelos elementares, que constituem hoje uma importante ferramenta no estudo de topologia geral. / The concept of reflection in topology expresses the fact that a space satisfies a given property provided that its \"small\" subspaces do. This work presents a study on reflection of properties concerning most of the main cardinal functions and metrizability, as well as other related properties. Open problems --such as Hamburger\'s question-- are also discussed, including partial answers and consistent examples. Several of the proofs presented here make use of elementary submodels, nowadays an important tool in the study of general topology.

O conceito de reflexão em topologia expressa o fato de que um espaço satisfaz uma dada propriedade sempre que esta é satisfeita por seus subespaços \"menores\". Neste trabalho, estuda-se a reflexão de propriedades envolvendo a maioria das principais funções cardinais e metrizabilidade, bem como outras propriedades relacionadas. São discutidos problemas em aberto -- como o problema de Hamburger --, incluindo respostas parciais e exemplos de consistência. Várias dentre as demonstrações apresentadas utilizam técnicas de submodelos elementares, que constituem hoje uma importante ferramenta no estudo de topologia geral. / The concept of reflection in topology expresses the fact that a space satisfies a given property provided that its \"small\" subspaces do. This work presents a study on reflection of properties concerning most of the main cardinal functions and metrizability, as well as other related properties. Open problems --such as Hamburger\'s question-- are also discussed, including partial answers and consistent examples. Several of the proofs presented here make use of elementary submodels, nowadays an important tool in the study of general topology.

In Riemannian geometry, the fundamental fact is that there exists a unique torsion-free connection (called the Levi-Civita connection) compatible with the Riemannian metric g, i.e. having the property ∇g = 0. In projective geometry, the class of covariant derivatives defining the geometry is fixed and all these covariant derivatives have the same class of (non- parametrized) geodesics. Old (and non-trivial) problem is to find whether these curves are geodesics of a (pseudo-)Riemannian metric. Such projective structures are called metrizable. Surprisingly enough, U. Dini and R. Liu- oville found in 19th century that the metrizability problem leads to a system of linear PDE's. In the last years, there were several papers dealing with these problems. The projective geometry is a representative example of the so called parabolic geometries (for full description, see the recent monograph by A. Čap and J. Slovák). It was realized recently that the corresponding linear metrizability operator is a special example of the so called first BGG operator. The flat model of projective geometry is the (real) projective space. In this more general context, the metrizability problem for (pseudo- )Riemannian geometries is naturally generalized to the sub-Riemannian situation. In the recent preprint, D.Calderbank, J....

This dissertation deals with three topics in descriptive set theory. First, the order topology is a natural topology on ordinals. In Chapter 2, a complete classification of order topologies on ordinals up to Borel isomorphism is given, answering a question of Benedikt Löwe. Second, a map between separable metrizable spaces X and Y preserves complete metrizability if Y is completely metrizable whenever X is; the map is resolvable if the image of every open (closed) set in X is resolvable in Y. In Chapter 3, it is proven that resolvable maps preserve complete metrizability, generalizing results of Sierpiński, Vainštein, and Ostrovsky. Third, an equivalence relation on a Polish space has the Laczkovich-Komjáth property if the following holds: for every sequence of analytic sets such that the limit superior along any infinite set of indices meets uncountably many equivalence classes, there is an infinite subsequence such that the intersection of these sets contains a perfect set of pairwise inequivalent elements. In Chapter 4, it is shown that every coanalytic equivalence relation has the Laczkovich-Komjáth property, extending a theorem of Balcerzak and Głąb.