On Some Aspects of Topology

Abstract

A topological space is a non-empty set X together with a collection T of subsets of X satisfying the conditions: (i) X, T, (ii) the union of any class of sets in T belongs to T, (iii) The intersection of a finite number of sets belongs to T, T is called a topology on X. The thesis is a study of several variants of topology obtained by generalizing some its aspects, viz, the conditions (ii) and (iii). The variants which have been considered here are the following: (1) a U-structure, (a topology in which the condition (iii) is omitted), (2) An I-structure, (a topology in which the condition (ii) is omitted), (3) A CU-structure, (a U-structure in which ‘any class’ in (ii) is replaced by ‘a countable class’), (4) A CUI-structure, (a topology in which ‘any class’ in (ii) is replaced by ‘a countable class’ ), (5) an FU-structure, (a U-structure in which ‘any class’ in (ii) is replaced by ‘a finite class’), (6) An FUI-structure, (a topology in which ‘any class’ in (ii) is replaced by ‘a finite class’). X together with the above structures (1) - (6) have been called a U- space, an I-space, a CU-space, a CUI-space, an FU-space and a FUI-space respectively. Among these, U-spaces and I-spaces have been defined and studied earlier by others and have been called supratopological spaces and infratopological spaces respectively. Our studies of these spaces in this thesis have considerably larger breadth and depth. The thesis has been divided into seven chapters. The first six chapters give detailed study of general properties, different kinds of compactness and compactification, several kinds of connectedness, various separation properties, projectives in some categories of U-spaces. The U-space version of most of the well-known and the important theorems for topological spaces have been proved to be valid. Very many suitable examples and counter examples have been constructed. In the last chapter the other kinds of the above-mentioned spaces have been dealt with. A few properties have been established and a few examples have been provided. Most of the properties proved for U-spaces do hold for these spaces as well. But these have not been stated and proved to avoid monotony or repetitions.