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Chapter 4 Hewitt-Nachbin Completeness and Continuous Mappings

DOI: 10.1016/s0304-0208(08)70849-1
  • Mathematics


Publisher Summary A topological property P is said to be invariant (respectively, inverse invariant) under a mapping f, if the image (respectively, inverse image) under f of a space with property P also has property P. This chapter discusses the invariance and inverse invariance of Hewitt–Nachbin completeness under various classes of continuous mappings. Unlike the property of compactness, the continuous image of a Hewitt–Nachbin space need not be Hewitt–Nachbin complete. An example is provided in the chapter showing that such is not the case even if the mapping happens to be a perfect mapping (also called a “proper mapping” or a “fitting mapping”). Hewitt–Nachbin completeness is invariant and inverse in variant under a perfect mapping whenever the domain is also normal and countably paracompact. Every compact space is paracompact and every paracompact Hausdorff space of nonmeasurable cardinal is Hewitt–Nachbin complete. If a perfect map is defined as a continuous closed surjection for which the inverse images of points are compact, then it is well known that compactness is both invariant and inverse invariant under perfect mappings.

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