Another generalization of the m-topology

Copyright c ○ 2014 Farshid Manshoor and Farhad Manshoor. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. It is well known that the mI-topology is a generalization of the m-topology on C(X), see [1]. Given two subsets A, B ⊆ X such that A ∪ B = X, we are going to define a topology on C(X) namely the m (A,B)-topology, finer than the m-topology and C(X) with this topology becomes a topological ring. Connectedness in this space is studied and it is shown that if A, B are closed realcompact subsets of X, then the component of the zero function in C(X) with m (A,B)-topology is the ideal CK(X).