In this paper we develop a point-free approach to the study of topological spaces and functions on them, establish platforms for both and present some findings on recursive points. (The effectivization of the functions on our spaces and related results are presented in a sequel.) In the first sections of the paper, we obtain conditions under which our approach leads to the generation of ideal objects (points) with which mathematicians work. Next, we apply the effective version of our approach to the real numbers, and make exact connections to the classical approach to recursive reals. In the succeeding sections of the paper, we introduce machinery to produce functions on topological spaces and find succinct conditions which will be effectivized in our sequel.
