#### Title

Characterizing topological properties by real functions

#### Publication Date

1975

#### Document Type

Conference Proceeding

#### Abstract

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.

## Comments

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