Characterizing topological properties by real functions

Publication Date


Document Type

Conference Proceeding


Guthrie JA, Henry M. Characterizing topological properties by real functions. 1975 1975:189-96.


This chapter discusses what completely regular space (X) can be characterized by the fact that some, or all, of the members of C(X) (collection of all continuous real-valued mappings defined on X) satisfy a given property. The requirement that X be completely regular is included to assure the existence of nonconstant members in C(X). It is well known that some of the most interesting classes of spaces will provide answers to this question as it is true that (1) X is compact if each fC(X) is perfect; (2) X is countably compact if each fC(X) is closed and a priori; and (c) X is pseudocompact if each fC(X) is bounded. The chapter shows that this list can be extended to include the first countable spaces, locally compact spaces, and the spaces of point-countable type, which are a common generalization of both.