On Gleason spaces

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Saudi Digital Library
The object of prime interest in this thesis is the projective resolutions in the category of compact Hausdorff spaces and continuous maps. However, related topics are also discussed both for their connection with the subject and for their intrinsic value. Chapter I serve as an introduction containing most of the fundamental concepts and techniques used in subsequent chapters. Four main topics are discussed: lattices and Boolean algebras, completely regular spaces, convergence of z-filters and z-ultrafilters, and zero-dimensional spaces. This material appears in the works of Dwinger [7], Gillman and Jerison [8], Halmos [12], and Walker [19]. Chapter II deals with the development of the Stone-Cech compactification and some of its basic properties. Some characterizations culminate in Glicksberg’s theorem on the Stone-Cech compactification of a product of two spaces. The discussion in this chapter is based on that of Bagley, Connell and McKnight [1], Gillman and Jerison [8], Glicksberg [9, 10], and Hewitt [13]. The necessary material on Boolean algebras, including the Stone representation theorem is discussed in Chapter III. The maximal ring of quotients of a commutative semisimple ring is given self-contained treatment and proves to be useful in connection with maximal ideal spaces and projective resolutions. Parts of this material are treated in the works of Brainerd and Lambek [4], Doctor [5], Lambek [15], Utumi [18], and Walker [19]. Chapter IV places major emphasis on the existence, uniqueness, and characterization of projective resolutions. This topic relates the Stone-Cech compactification to both Boolean algebras and the class of extremally disconnected spaces. Theorem 3 in this chapter is a new result on the product of Gleason spaces while Corollary 2 provides a new simple proof of the fact that a projective object is extremally disconnected. The discussion in this chapter is based on the works of Banaschewski [2, 3], Hager [11], Park [16], and Rainwater [17].