Connectedness and compactness in C(X) with the m-topology and generalized m-topology

We give a generalization of the m-topology on C(X) and investigate the connectedness and compactness in C(X) with this topology. Using this, it turns out that compact subsets in Cm(X) (C(X) with the m-topology) have empty interior and an ideal in Cm(X) is connected if and only if it is contained in every hyper-real maximal ideal of C(X). We show that the component of 0 in Cm(X) is Cψ(X), the set of all functions in C(X) with pseudocompact support. It is also shown that the components and the quasicomponents in Cm(X) coincide. Topological spaces X are characterized for which Cm(X) is connected, locally connected or totally disconnected. We observe that locally compactness, σ-compactness and hemicompactness of Cm(X) are all equivalent to X being finite. Finally, we have shown that if M is a maximal ideal in C(X), then C(X)/M with the m-topology is connected if and only if M is real.