Cofinite Topological Manifolds And Invariance Of Topological Properties With Respect To Almost Continuous Functions

ABSTRACT

Manifolds are generalization of curves and surfaces to arbitrary higher dimensions. They are of many kinds, one of them being topological manifolds. The main feature common to manifolds is that every point of the space is in one to one correspondence with a point in another space. Hausdorff manifolds have been developed on infinite dimensional spaces such as Banach spaces and Fréchet spaces. Topological properties of non-Hausdorff manifolds have been studied and the notion of compatible apparition points have been introduced for such non-Hausdorff manifolds. Among many generalizations of the notion of continuity, almost continuous functions have been used on both 1 T and 2 T  spaces. In such spaces, the properties and characterization of almost continuous functions have been studied and interesting results have been obtained especially with the class of 1 T  spaces. Invariance and inverse invariance of some topological properties have been investigated with respect to almost continuous functions and continuous functions. Manifolds have not been modeled on a 1 T  space having cofinite topology. Invariance of topological properties such as compactness and its other notions have not been investigated from an infinite cofinite space to the Euclidean space with respect to almost continuous functions. This study has therefore investigated the invariance of these topological properties from the cofinite space to the Euclidean ℝ n spaces with respect to almost continuous functions. Differentiable manifolds are among the most fundamental notions of modern mathematics as it is the cornerstone of modern mathematical science. The study has also obtained a cofinite manifold where almost continuous functions have been used as maps. Pseudoderivative on almost continuous functions has also been defined in this study and some of its properties have been stated. This has facilitated the development of pseudodifferentiable cofinite manifolds. Manifolds, especially the differentiable manifolds have applications in survey and physics. In physics, the applications are found in mechanics and electromagnetics where boundary value problems are solved in mesh generation.