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Kleene’s computability theory based on the S1–S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing’s ‘machine model’ which formalises computing with real numbers. A fundamental distinction in Kleene’s framework is between normal and non-normal functionals where the former compute the associated Kleene quantifier ∃ n and the latter do not. Historically, the focus was on normal functionals, but recently new non-normal functionals have been studied based on well-known theorems, the weakest among which seems to be the uncountability of the reals. These new non-normal functionals are fundamentally different from historical examples like Tait’s fan functional: the latter is computable from ∃ 2 , while the former are computable in ∃ 3 but not in weaker oracles. Of course, there is a great divide or abyss separating ∃ 2 and ∃ 3 and we identify slight variations of our new non-normal functionals that are again computable in ∃ 2 , i.e. fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, bounded variation, and semi-continuity from real analysis.
Kleene’s computability theory based on the S1–S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing’s ‘machine model’ which formalises computing with real numbers. A fundamental distinction in Kleene’s framework is between normal and non-normal functionals where the former compute the associated Kleene quantifier ∃ n and the latter do not. Historically, the focus was on normal functionals, but recently new non-normal functionals have been studied based on well-known theorems, the weakest among which seems to be the uncountability of the reals. These new non-normal functionals are fundamentally different from historical examples like Tait’s fan functional: the latter is computable from ∃ 2 , while the former are computable in ∃ 3 but not in weaker oracles. Of course, there is a great divide or abyss separating ∃ 2 and ∃ 3 and we identify slight variations of our new non-normal functionals that are again computable in ∃ 2 , i.e. fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, bounded variation, and semi-continuity from real analysis.
<abstract><p>In this paper, $ \omega _{s} $-irresoluteness as a strong form of $ \omega _{s} $-continuity is introduced. It is proved that $ \omega _{s} $-irresoluteness is independent of each of continuity and irresoluteness. Also, $ \omega _{s} $-openness which lies strictly between openness and semi-openness is defined. Sufficient conditions for the equivalence between $ \omega _{s} $-openness and openness, and between $ \omega _{s} $-openness and semi-openness are given. Moreover, pre-$ \omega _{s} $ -openness which is a strong form of $ \omega _{s} $-openness and independent of each of openness and pre-semi-openness is introduced. Furthermore, slight $ \omega _{s} $-continuity as a new class of functions which lies between slight continuity and slight semi-continuity is introduced. Several results related to slight $ \omega _{s} $-continuity are introduced, in particular, sufficient conditions for the equivalence between slight $ \omega _{s} $ -continuity and slight continuity, and between slight $ \omega _{s} $ -continuity and slight semi-continuity are given. In addition to these, $ \omega _{s} $-compactness as a new class of topological spaces that lies strictly between compactness and semi-compactness is introduced. It is proved that locally countable compact topological spaces are $ \omega _{s} $ -compact. Also, it is proved that anti-locally countable $ \omega _{s} $ -compact topological spaces are semi-compact. Several implications, examples, counter-examples, characterizations, and mapping theorems are introduced related to the above concepts are introduced.</p></abstract>
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