Philosophiae Doctor - PhD / Categorical closure operators were introduced by Dikranjan and Giuli in [DG87] and then developed by these authors and Tholen in [DGT89]. These operators have played an important role in the development of Categorical Topology by introducing topological concepts, such as connectedness, separatedness and compactness, in an arbitrary category and they provide a uni ed approach to various mathematical notions. Motivated by the theory of these operators, the categorical notion of interior operators was introduced by Vorster in [Vor00]. While there is a notational symmetry between categorical closure and interior operators, a detailed analysis shows that the two operators are not categorically dual to each other, that is: it is not true in general that whatever one does with respect to closure operators may be done relative to interior operators. Indeed, the continuity condition of categorical closure operators can be expressed in terms of images or equivalently, preimages, in the same way as the usual topological closure describes continuity in terms of images or preimages along continuous maps. However, unlike the case of categorical closure operators, the continuity condition of categorical interior operators can not be described in terms of images. Consequently, the general theory of categorical interior operators is not equivalent to the one of closure operators. Moreover, the categorical dual closure operator introduced in [DT15] does not lead to interior operators. As a consequence, the study of categorical interior operators in their own right is interesting.

Philosophiae Doctor - PhD / In a category C with a proper (E; M)-factorization system for morphisms, we further investigate categorical topogenous structures and demonstrate their prominent role played in providing a uni ed approach to the theory of closure, interior and neighbourhood operators. We then introduce and study an abstract notion of C asz ar's syntopogenous structure which provides a convenient setting to investigate a quasi-uniformity on a category. We demonstrate that a quasi-uniformity is a family of categorical closure operators. In particular, it is shown that every idempotent closure operator is a base for a quasi-uniformity. This leads us to prove that for any idempotent closure operator c (interior i) on C there is at least a transitive quasi-uniformity U on C compatible with c (i). Various notions of completeness of objects and precompactness with respect to the quasi-uniformity de ned in a natural way are studied. The great relationship between quasi-uniformities and closure operators in a category inspires the investigation of categorical quasi-uniform structures induced by functors. We introduce the continuity of a C-morphism with respect to two syntopogenous structures (in particular with respect to two quasi-uniformities) and utilize it to investigate the quasiuniformities induced by pointed and copointed endofunctors. Amongst other things, it is shown that every quasi-uniformity on a re ective subcategory of C can be lifted to a coarsest quasi-uniformity on C for which every re ection morphism is continuous. The notion of continuity of functors between categories endowed with xed quasi-uniform structures is also introduced and used to describe the quasi-uniform structures induced by an M- bration and a functor having a right adjoint.

The focus of this research is the extension of topological operators with the addition of a inclusion measure. This extension is carried out in both crisp and fuzzy topological spaces. The mathematical properties of the new operators are discussed and compared with traditional operators. Ignoring small errors due to imperfections and noise in digital images is the main motivation in introducing the proposed operators. To show the effectiveness of the new operators, we demonstrate their utility in image database classification and shape classification. Each image (shape) category is modeled with a topological space and the interior of the query image is obtained with respect to different topologies. This novel way of looking at the image categories and classifying a query image shows some promising results. Moreover, the proposed interior and closure operators with inclusion degree is utilized in mathematical morphology area. The morphological operators with inclusion degree outperform traditional morphology in noise removal and edge detection in a noisy environment

The focus of this research is the extension of topological operators with the addition of a inclusion measure. This extension is carried out in both crisp and fuzzy topological spaces. The mathematical properties of the new operators are discussed and compared with traditional operators. Ignoring small errors due to imperfections and noise in digital images is the main motivation in introducing the proposed operators. To show the effectiveness of the new operators, we demonstrate their utility in image database classification and shape classification. Each image (shape) category is modeled with a topological space and the interior of the query image is obtained with respect to different topologies. This novel way of looking at the image categories and classifying a query image shows some promising results. Moreover, the proposed interior and closure operators with inclusion degree is utilized in mathematical morphology area. The morphological operators with inclusion degree outperform traditional morphology in noise removal and edge detection in a noisy environment