Connectedness and compactness in 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 C m ( X ) ( C ( X ) with the m-topology) have empty interior and an ideal in C m ( 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 C m ( X ) is C ψ ( X ) , the set of all functions in C ( X ) with pseudocompact support. It is also shown that the components and the quasicomponents in C m ( X ) coincide. Topological spaces X are characterized for which C m ( X ) is connected, locally connected or totally disconnected. We observe that locally compactness, σ-compactness and hemicompactness of C m ( 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.