Abstract
This chapter presents a formal connection between the notions of projective spaces and projective resolutions for compact and noncompact completely regular spaces. In the category of compact, completely regular spaces projective spaces are assumed to be projective objects in regard to the class A of all mappings onto. The class of all projective spaces of this category is equal to each of the class of spaces: (1) all extremally disconnected spaces, (2) all spaces each mapping onto which is a retraction. In the category of completely regular spaces, to exclude the triviality (= discreteness), projective spaces are assumed to be projective objects in regard to the class A of all perfect mappings onto, that is, closed mappings onto, for which all inverse images of points are compact.
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