Connectedness and compactness in \(C(X)\) with the \(m\)-topology and generalized \(m\)-topology
For a completely regular Hausdorff space \(X\), let \(C_{m}(X)\) be the ring of all real-valued continuous functions on \(X\) endowed with the \(m\)-topology. In this paper, the authors give a generalization of the \(m\)-topology on \(C(X)\) and investigate connectedness and compactness in \(C(X)\) with this topology. Using this, it turns out that compact subsets in \(C_{m}(X)\) 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 \(C(X)\). The authors show that the component of 0 in \(C_{m}(X)\) is \(C_{\psi}(X)\), 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. Furthermore, topological spaces \(X\) are characterized for which \(C_{m}(X)\) is connected, locally connected or totally disconnected. The authors obtain that local compactness, \(\sigma\)-compactness and hemicompactness of \(C_{m}(X)\) are all equivalent to \(X\) being finite. Examples of Lindelöf and non-Lindelöf ideals in \(C_{m}(X)\) are given and it is shown that the Lindelöf ideals of \(C_{m}(X)\) are also contained in \(C_{\psi}(X)\).
totally disconnected and hyper-real maximal ideal
scientific article; zbMATH DE number 6092993