© 1996, Topology Atlas
A subset B of a Tychonoff space X is said to be functionally bounded if every real-valued continuous f defined on X is bounded on B (i.e. f(B) is a bounded subset of R). Functionally bounded subsets and some related concepts arose, in a natural way, during the study of the topological properties induced by real-valued continuous functions. Functionally boundedness generalizes pseudocompactness: A space is pseudocompact if it is functionally bounded in itself. This concept was implicit in the well-known theorem of Nachbin-Shirota which characterizes when the space of real-valued continuous functions endowed with the compact-open topology is barrelled. The above definition appears in a paper of T.Isiwata (1967) (he called these subsets relatively pseudocompact) related with the theory of Z-mappings, WZ-mappings and extension of open mappings. In the context of the theory of the bounded subsets of locally convex spaces, H.Buchwalter (1969) called these subsets bounded. The denomination functionally bounded has been used in the theory of Hausdorff topological groups (J.Trigos-Arrieta, 1991) to distinguish between this concept and the concept of the bounded subset in the sense of precompact subset.
Funtionally boundedness has been applied in differents contexts, for instance: nearly realcompact spaces, Grothendieck's theorems, spaces with an OZ Stone-Cech compactification, in the problem of the distribution of the functor of the realcompactification of Hewitt and the topological completion, Ascoli's theorems, continuous function spaces, countably compactifications (in the sense of Morita), etc.
There are several important kinds of functionally bounded subsets. For instance, we can cite:
(1) strongly bounded subsets, (M.Tkachenko, 1988): subsets such that its product by every functionally bounded subset is also functionally bounded. These subsets play an important role in the field of Hausdorff topological groups: every functionally bounded subset of a Hausdorff topological group is strongly bounded.
(2) C_alpha-compact subsets: a subset B of X is C_alpha-compact if the restriction to B of every continuous function from X into R^alpha has compact range. C_alpha-compactness generalizes alpha-pseudocompacness of Kennison (1961). The case alpha = omega_0 was introduced by T.Isiwata (1967) under the name "subsets with property (*)". H.Buchwalter (1969) called them hyperbounded and R.L.Blair and M.A. Swardson (1990) strongly relatively pseudocompact subsets. The denomination C-compact is due to S.Garcia-Ferreira and A.Garcia-Maynez. These last authors related this concept with the concept of weakly-pseudocompact spaces.
At present, there are several important questions related to this class of functionally bounded subsets: