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Motivated by the two remarks, that the study of computability based on theuse of numberings -- Type 1 computability -- does not have to be restricted tocountable sets equipped with onto numberings, and that computable topologieshave been in part developed with the implicit hypothesis that the consideredspaces should be computably separable, we propose new definitions for Type 1computable topological spaces. We define computable topological spaces withoutmaking reference to a basis. Following Spreen, we show that the use of a formalinclusion relation should be systematized, and that it cannot be avoided if wewant computable topological spaces to generalize computable metric spaces. Wealso compare different notions of effective bases. The first one, introduced byNogina, is based on an effective version of the statement "a set $O$ is open iffor any point in $O$, there is a basic set containing that point and containedin $O$''. The second one, associated to Lacombe, is based on an effectiveversion of "a set $O$ is open if it can be written as a union of basic opensets''. We show that neither of these notions of basis is completelysatisfactory: Nogina bases do not permit to define computable topologies unlesswe restrict our attention to countable sets, and the conditions associated toLacombe bases are too restrictive, and they do not apply to metric spacesunless we add effective separability hypotheses. We define a new notion ofbasis, based on an effective version of the Nogina statement, but adding to itseveral classically empty conditions, expressed in terms of formal inclusionrelations. Finally, we obtain a new version of the theorem of Moschovakis whichstates that the Lacombe and Nogina approaches coincide on countable recursivePolish spaces, but which applies to sets equipped with non-onto numberings, andwith effective separability as a sole hypothesis.

New definitions in the theory of Type 1 computable topological spaces