A generalization of the m-topology on C(X) finer than the m-topology

Abstract

It is well known that the component of the zero function in C(X) with the m-topology is the ideal C ψ (X). Given any ideal I ⊆ C ψ (X), we are going to define a topology on C(X) namely the m I-topology, finer than the m-topology in which the component of 0 is exactly the ideal I and C(X) with this topology becomes a topological ring. We show that compact sets in C(X) with the m I-topology have empty interior if and only if X \ Z[I] is infinite. We also show that nonzero ideals are never compact, the ideal I may be locally compact in C(X) with the m I-topology and every Lindelöf ideal in this space is contained in C ψ (X). Finally, we give some relations between topological properties of the spaces X and C m (X). For instance, we show that the set of units is dense in C m (X) if and only if X is strongly zero-dimensional and we characterize the space X for which the set r(X) of regular elements of C(X) is dense in C m (X).