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Approximating Topological Spaces by Polyhedra

Elsevier B.V.
DOI: 10.1016/b978-044451861-3/50006-9
  • Mathematics


Publisher Summary This chapter discusses the problem of approximating topological spaces by polyhedra using various techniques based on inverse systems. The chapter outlines the properties preserved under inverse limits and states theorems concerning the representation of compact Hausdorff spaces as limits of inverse systems with desired additional properties. One of the ultimate goals of topology is to understand the structure of topological spaces and (continuous) mappings. Every topologically complete space X is the limit of an inverse system of poly hedra. It is said that a property C of spaces is inherited by the inverse limit, or equivalently, a class of spaces C is closed under inverse limits, provided XλɛC implies X ɛC. The study of a space X having a certain property C is greatly facilitated if one can represent X as the limit of an inverse system of polyhedra Xλ having property C. The notion of an inverse limit looses most of its good properties when one abandons the realm of compact spaces.

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