Neste trabalho, introduzimos uma generalização da noção de módulo de homologia local de um módulo com respeito a um ideal, o qual nós chamamos de módulo de homologia local com respeito a um par de ideais. Estudamos suas várias propriedades tais como teoremas de anulamento e de não anulamento, e Artinianidade. Também fazemos sua conexão com a homologia e cohomologia local usual. Introduzimos uma generalização da noção de largura de um ideal sobre um módulo aplicando o conceito de módulo de homologia local com respeito a um par de ideais. Também introduzimos o conceito de um módulo co-Cohen-Macaulay para um par de ideais, o qual é uma generalização o conceito de um módulo co-Cohen-Macaulay. Para finalizar, introduzimos o limite inverso de homologia local, e estudamos algumas de suas propriedades, analisamos a sua estrutura, o anulamento, não anulamento e Artinianidade. / In this work, we introduce a generalization of the notion of local homology module of a module with respect to an ideal, which we call of local homology module with respect to a pair of ideals. We study its various properties such as vanishing and nonvanishing theorems, and Artinianness. We also do its connection with ordinary local homology and cohomology. We introduce a generalization of the notion of width of an ideal on a module applying the concept of local homology module with respect to a pair of ideals. Also we introduce the concept of a co-Cohen-Macaulay module for a pair of ideals, what is a generalization of the concept of a co-Cohen-Macaulay module. To finish, we introduce the inverse limit of local homology, and we study some of its properties, we analyze the their structure, the vanishing, non-vanishing and Artinianness.

In part one we show that for a compact, metric, locally path-connected topological space X, the shape group of X - as defined in Foundations of Shape Theory by Mardesic and Segal - is isomorphic to the inverse limit of discrete homotopy groups introduced by Conrad Plaut and Valera Berestovskii. We begin by providing the reader preliminary definitions of the fundamental group of a topological space, inverse systems and inverse limits, the Shape Category, discrete homotopy groups, and culminate by providing an isomorphism of the shape and deck groups for peano continua. In part two we develop work and provide further classification of Sharkovskii topological groups, which we call Sharkovskii Groups. We culminate in proving the fact that a locally compact Sharkovskii group must either be the real numbers if it is not compact, or a torsion-free solenoid if it is compact.

Let H be the Hawaiian Earring, and let H denote its fundamental group. Assume (Bi) is an inverse system of bouquets of circles whose inverse limit is H. We give an explicit bijection between finite normal covering spaces of H and finite normal covering spaces of Bi. This bijection induces a correspondence between a certain family of inverse sequences of these covering spaces. The correspondence preserves the inverse limit of these sequences, thus offering two methods of constructing the same limit. Finally, we characterize all spaces that can be obtained in this fashion as a particular type of fibrations of H.

We present a family of fibrations over the Hawaiian earring that are inverse limits of regular covering spaces over the Hawaiian earring. These fibrations satisfy unique path lifting, and as such serve as a good extension of covering space theory in the case of nonsemi-locally simply connected spaces. We give a condition for when these fibrations are path-connected.