Fredholm separating maps on continuous function spaces

For locally compact Hausdorff spaces X and Y, we initiate a study of Fredholm separating maps T from C(X) into C(Y). Our key aim is to completely recognize the structures of the kernel space and the range space of T in terms of merging points to show that X and Y are homeomorphic after removing related finite subsets; as the main result, we describe the interaction between continuity of T and closedness of its range; in particular, under certain condition, we conclude that T can be represented as a weighted composition operator.