Affordable Access

Publisher Website

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.

There are no comments yet on this publication. Be the first to share your thoughts.