The m-topology on Cm(X) revisited

Hewitt [E. Hewitt, Rings of real-valued continuous functions, I, Trans. Amer. Math. Soc. 64 (1948) 45–99], generalizing work of E.H. Moore, defined the m-topology on C(X). In his article he demonstrated that certain classes of topological spaces X can be characterized by topological properties of Cm(X). For example, he showed that X is pseudocompact if and only if Cm(X) is first countable. Others have also investigated topological properties of X via properties of Cm(X), e.g., [G. Di Maio, L. Holá, D. Holý, R.A. McCoy, Topologies on the space of continuous functions, Topology Appl. 86 (2) (1998) 105–122] and [E. van Douwen, Nonnormality or hereditary paracompactness of some spaces of real functions, Topology Appl. 39 (1) (1991) 3–32]. We continue this practice in the second section and give some new equivalent characterizations. In the third section we prove the converse of a theorem of van Douwen [E. van Douwen, Nonnormality or hereditary paracompactness of some spaces of real functions, Topology Appl. 39 (1) (1991) 3–32] completing a characterization of when Cm(X) is a weak P-space. In the fourth section we determine when Cm(X) has no non-trivial convergent sequences.