Dyadic Subbases and Representations of Topological Spaces

We explain topological properties of the embedding-based approach to computability on topological spaces [10–12]. With this approach, a special kind of embeddings of topological spaces into Plotkin’s Tω, which is the set of infinite sequences of T = {0, 1,⊥}, are considered. Such an embedding can also be characterized by a dyadic subbase, which is a countable subbase S = (S 0 , S 1 0 , S 0 1 , S 1 1 , . . .) such that S n(n = 0, 1, 2, . . . , j = 0, 1) are regular open and S 0 n and S 1 n are exteriors of each other. We survey, based on [12], properties of dyadic subbases which are related to efficiency properties of the representation corresponding to the embedding.