This paper is a study of continue and related metric spaces, Chapter I is an introductory chapter. Irreducible continua and noncut points are the main topics in Chapter II. The third chapter begins with a few results on locally connected spaces. These results are then used to prove results in locally connected continua. Decomposable and indecomposable continua are dealt with in Chapter IV. Totally disconnected metric spaces are studied in the beginning of Chapter V. Then we see that every compact metric space is a continuous image of the Cantor set. A continuous map from the Cantor set onto [0,1] is constructed. Also, a continuous map from [0,1] onto [0,1]x[0,1] is built, Then an order preserving homeomorphism is constructed from a metric arc onto [0,1],

In this paper the weak completeness of certain sequence spaces is examined. In particular, we show that each of the sequence spaces c0 and 9, 1 < p < c, is a Banach space. A Riesz representation for the dual space of each of these sequence spaces is given. A Riesz representation theorem for Hilbert space is also proven. In the third chapter we conclude that any reflexive space is weakly (sequentially) complete. We give 01 as an example of a non-reflexive space that is weakly complete. Two examples, c0 and YJ, are given of spaces that fail to be weakly complete.

This study of classical and modern harmonic analysis extends the classical Wiener's approximation theorem to locally compact abelian groups. The first chapter deals with harmonic analysis on the n-dimensional Euclidean space. Included in this chapter are some properties of functions in L1(Rn) and T1(Rn), the Wiener-Levy theorem, and Wiener's approximation theorem. The second chapter introduces the notion of standard function algebra, cospectrum, and Wiener algebra. An abstract form of Wiener's approximation theorem and its generalization is obtained. The third chapter introduces the dual group of a locally compact abelian group, defines the Fourier transform of functions in L1(G), and establishes several properties of functions in L1(G) and T1(G). Wiener's approximation theorem and its generalization for L1(G) is established.

In this thesis we present the Stone representation theorem, generally known as Stone duality in the point-free context. The proof is choice-free and, since we do not have to be concerned with points, it is by far simpler than the original. For each infinite cardinal κ we show that the counter- part of the κ-complete Boolean algebras is constituted by the κ-basically disconnected Stone frames. We also present a precise characterization of the morphisms which correspond to the κ-complete Boolean homomorphisms. Although Booleanization is not functorial in general, in the part of the dual- ity for extremally disconnected Stone frames it is, and constitutes an equiv- alence of categories. We finish the thesis by focusing on the De Morgan (or extremally disconnected) frames and present a new characterization of these by their superdense sublocales. We also show that in contrast with this phenomenon, a metrizable frame has no non-trivial superdense sublocale; in other words, a non-trivial Čech-Stone compactification of a metrizable frame is never metrizable. 1