Abstract
This chapter explains the completions in algebra and topology. Localization and completion are among the fundamental and first tools in commutative algebra. They play a correspondingly fundamental role in algebraic topology. Localizations and completions of spaces and spectra are the central tools since the 1970s. The topological constructions require the foundations that deal with the algebraically familiar theory of localization at multiplicatively closed subsets. The chapter explains the deeper and less familiar theory of completion, together with an ideal theoretic variant of localization. There is a still more general theory of localization of spaces and spectra at spectra.
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