Characterizing topological properties by real functions
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 f ∈ C(X) is perfect; (2) X is countably compact if each f ∈ C(X) is closed and a priori; and (c) X is pseudocompact if each f ∈ C(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.