let f: R -> R be injective where R is given by the cofinite topology. Show that f is continuous.
so i know that the cofinite topology consists of sets that are either empty or have finite complements. in the case of R, the open sets in the topology are intervals containing + and - infinite while leaving out finitely many terms. to prove that f is continuous i have to show that given any open set V in the codomain, the preimage f^-1 (V) in the domain is also open. i also know that in order for a set to be open in R, there must exist a ball centered at each point and contained within the set for every point in that set.
i can see intuitively that since f is injective, the cardinality of V is less than or equal that of the preimage of V. and since V is open the preimage of V should be open as well. however that was based on my intuition and probably does not constitute a formal proof. how would you show that the preimage of any V is open?