## Continuous Functionals of Dependent and Transfinite Types (1995)

Citations: | 10 - 2 self |

### BibTeX

@MISC{Berger95continuousfunctionals,

author = {Ulrich Berger},

title = {Continuous Functionals of Dependent and Transfinite Types},

year = {1995}

}

### OpenURL

### Abstract

this paper we study some extensions of the Kleene-Kreisel continuous functionals [7, 8] and show that most of the constructions and results, in particular the crucial density theorem, carry over from nite to dependent and transnite types. Following an approach of Ershov we dene the continuous functionals as the total elements in a hierarchy of Ershov-Scott-domains of partial continuous functionals. In this setting the density theorem says that the total functionals are topologically dense in the partial ones, i.e. every nite (compact) functional has a total extension. We will extend this theorem from function spaces to dependent products and sums and universes. The key to the proof is the introduction of a suitable notion of density and associated with it a notion of co-density for dependent domains with totality. We show that the universe obtained by closing a given family of basic domains with totality under some quantiers has a dense and co-dense totality provided the totalities on the basic domains are dense and co-dense and the quantiers preserve density and co-density. In particular we can show that the quantiers and have this preservation property and hence, for example, the closure of the integers and the booleans (which are dense and co-dense) under and has a dense and co-dense totality. We also discuss extensions of the density theorem to iterated universes, i.e. universes closed under universe operators. From our results we derive a dependent continuous choice principle and a simple order-theoretic characterization of extensional equality for total objects. Finally we survey two further applications of density: Waagb's extension of the Kreisel-Lacombe-Shoeneld-Theorem showing the coincidence of the hereditarily eectively continuous hierarchy...