This thesis presents a development of some useful concepts concerning topological spaces. Most of the theorems given apply to the most general form of topological space.

One of the classic theorems concerning the real numbers states that every open cover of a closed and bounded subset of the real line contains a finite subcover. Compactness is an abstraction of that notion, and there are several ideas concerning it which are equivalent and many which are similar. The purpose of this paper is to synthesize the more important of these ideas. This synthesis is accomplished by demonstrating either situations in which two ordinarily different conditions are equivalent or combinations of two or more properties which will guarantee a third.

The need of finding a way to design a company trademark, without the worry of possible infringement on the intellectual property rights, has become exceedingly important as the economy and the accompanying intellectual property concerns advanced greatly in recent years. Traditionally, registered trademarks are stored in image databases and are categorized and retrieved by descriptions and keywords given by human workers. This is extremely time-consuming and considered by many as inappropriate. In this work we focus on image feature and content related techniques, or content-based image retrieval (CBIR) methods. Nevertheless, we still need human inputs since by law the most crucial basis for discerning the similarity or difference of two trademarks has to rely on human¡¦s naked eye. Therefore in this work we created a program which incorporates an man-machine interface allowing users to input various weighting factors each emphasizing a specific feature or shape of the trademark. The Zernike moments, and some new image compactness indices are used in the computations for image comparisons.

This thesis covers fundamental properties of metric spaces, as well as completeness, compactness, and separability of metric spaces.

Master’s thesis is devoted to the study of cardinal invariants in the F-compact spaces class. Here and throughout the paper, the concept ”compact” would mean a compact Hausdorff space. In my thesis I have tried to present and explain all necessary concepts and statements necessary for the reader to get acquainted with F-compact spaces class. In order to understand the idea of F-compact spaces, it is necessary to understand what the inverse spectrum is from itself, it is necessary to know about the cardinality of sets and to understand that two infinite sets can have different cardinalities, know about closed and open sets, and much else that you will find in this paper. In the thesis the analysis of the scientific literature sources is presented; the theorems about the relationship between the characteristics of cardinality invariants in the F-compact spaces class are investigated; the relationships between the properties of perfect normality and hereditary normality in the F - compact spaces class of countable spectral height are studied. In the process of the investigation some propositions were found, proved and filled in the missing fragments of evidence. Conclusion: At present, the method of fully closed mappings (which is used in constructing of F - compact spaces ) is the most productive method of constructing counterexamples in general topology. I believe, that this paper will be interesting to all who wants to go beyond the ordinary, habitual way of thinking, because only by studying topology we can speak clearly and precisely about things related to the idea of continuity and infinity!

For smooth bounded pseudoconvex domains in Cn, we provide geometric conditions on (the points of infinite type in) the boundary which imply compactness of the ∂-Neumann operator. This is an extension of a theorem of Straube for smooth bounded pseudoconvex domains in C2.

Thesis (M.Sc. (Mathematics)) -- University of Limpopo, 2022 / Our discussion starts with the study of convergence and clustering of filters initiated in pointfree setting by Hong, and then characterize compact and almost compact frames in terms of these filters. We consider the strict extension and show that tQL is a zerodimensional compact frame, where Q denotes the set of filters in L. Furthermore, we study the notion of general filters introduced by Banaschewski and characterize compact frames and almost compact frames using them. For filter selections, we consider F−compact and strongly F−compact frames and show that lax retracts of strongly F−compact frames are also strongly F−compact. We study further the ideals Rs(L) and RK(L) of the ring of realvalued continuous functions on L, RL. We show that Rs(L) and RK(L) are improper ideals of RL if and only if L is compact. We consider also fixed ideals of RL and showthat L is compact if and only if every ideal of RL is fixed if and only if every maximalideal of RL is fixed. Of interest, we consider the class of isocompact locales, which is larger that the class of compact frames. We show that isocompactness is preserved by nearly perfect localic surjections. We study perfect compactifications and show that the Stone-Cˇech compactifications and Freudenthal compactifications of rim-compact frames are perfect. We close the discussion with a small section on Z−closed frames and show that a basically disconnected compact frame is Z−closed.

Abstract Not Provided / Thesis / Master of Science (MSc)

Gromov’s compactness theorem for metric spaces, a compactness theorem for the space of compact metric spaces equipped with the Gromov-Hausdorff distance, is a theorem with many applications. In this thesis, we give a generalisation of this landmark result, more precisely, we give a compactness theorem for the space of distance measure spaces equipped with the generalised Gromov-Hausdorff-Levi-Prokhorov distance. A distance measure space is a triple (X, d,µ), where (X, d) forms a distance space (a generalisation of a metric space where, we allow the distance between two points to be infinity) and µ is a finite Borel measure. Using this result we prove that the Deligne-Mumford compactification is the completion of the moduli space of Riemann surfaces under the generalised Gromov-Hausdorff-Levi-Prokhorov distance. The Deligne-Mumford compactification, a compactification of the moduli space of Riemann surfaces with explicit description of the limit points, and the closely related Gromov compactness theorem for J-holomorphic curves in symplectic manifolds (in particular curves in an algebraic variety) are important results for many areas of mathematics. While Gromov compactness theorem for J-holomorphic curves in symplectic manifolds, is an important tool in symplectic topology, its applicability is limited by the lack of general methods to construct pseudo-holomorphic curves. One hopes that considering a more general class of objects in place of pseudo-holomorphic curves will be useful. Generalising the domain of pseudo-holomorphic curves from Riemann surfaces to Riemann surface laminations is a natural choice. Theorems such as the uniformisation theorem for surface laminations by Alberto Candel (which is a partial generalisation of the uniformisation theorem for surfaces), generalisations of the Gauss-Bonnet theorem proved for some special cases, and topological classification of “almost all" leaves using harmonic measures reinforces the usefulness of this line on enquiry. Also, the success of essential laminations, as generalised incompressible surfaces, in the study of 3-manifolds suggests that a similar approach may be useful in symplectic topology. With this motivation, we prove a compactness theorem analogous to the Deligne-Mumford compactification for the space of Riemann surface laminations.

Hardy-type operators involving suprema have turned out to be a useful tool in the theory of interpolation, for deriving Sobolev-type inequalities, for estimates of the non-increasing rearrangements of fractional maximal functions or for the description of norms appearing in optimal Sobolev embeddings. This thesis deals with the compactness of these operators on weighted Banach function spaces. We de ne a category of pairs of weighted Banach function spaces and formulate and prove a criterion for the compactness of a Hardy-type operator involving supremum which acts between a couple of spaces belonging to this category. Further, we show that the category contains speci c pairs of weighted Lebesgue spaces determined by a relation between the exponents. Besides, we bring an extension of the criterion to all weighted Lebesgue spaces, in proof of which we use characterization of the compactness of operators having the range in the cone of non-negative non-increasing functions, introduced as a separate result.