In [G. Curi, "Exact approximations to Stone-Cech compactification'', Ann. Pure Appl. Logic, 146, 2-3, 2007, pp. 103-123] a characterization is obtained of the class of locales of which the Stone-Cech compactification can be defined in constructive type theory CTT, and in the formal system CZF+uREA+DC, a natural extension of Aczel's system for constructive set theory CZF by a strengthening of the Regular Extension Axiom REA and the principle of dependent choice. In this paper we show that this characterization continues to hold over the standard system CZF plus REA, thus removing in particular any dependency from a choice principle. This will follow by a result of independent interest, namely the proof that the class of continuous mappings from a compact regular locale X to a regular a set-presented locale Y is a set in CZF, even without REA. It is then shown that the existence of Stone-Cech compactification of a non-degenerate Boolean locale is independent of the axioms of CZF (+REA), so that the obtained characterization characterizes a proper subclass of locales. The same also holds for several, even impredicative, extensions of CZF+REA, as well as for CTT. This is in contrast with what happens in the context of Higher-order Heyting arithmetic HHA - and thus in any topos-theoretic universe: by constructions of Johnstone, Banaschewski and Mulvey, within HHA Stone-Cech compactification can be defined for every locale.
On the existence of Stone-Čech compactification.
CURI, Giovanni
In corso di stampa
Abstract
In [G. Curi, "Exact approximations to Stone-Cech compactification'', Ann. Pure Appl. Logic, 146, 2-3, 2007, pp. 103-123] a characterization is obtained of the class of locales of which the Stone-Cech compactification can be defined in constructive type theory CTT, and in the formal system CZF+uREA+DC, a natural extension of Aczel's system for constructive set theory CZF by a strengthening of the Regular Extension Axiom REA and the principle of dependent choice. In this paper we show that this characterization continues to hold over the standard system CZF plus REA, thus removing in particular any dependency from a choice principle. This will follow by a result of independent interest, namely the proof that the class of continuous mappings from a compact regular locale X to a regular a set-presented locale Y is a set in CZF, even without REA. It is then shown that the existence of Stone-Cech compactification of a non-degenerate Boolean locale is independent of the axioms of CZF (+REA), so that the obtained characterization characterizes a proper subclass of locales. The same also holds for several, even impredicative, extensions of CZF+REA, as well as for CTT. This is in contrast with what happens in the context of Higher-order Heyting arithmetic HHA - and thus in any topos-theoretic universe: by constructions of Johnstone, Banaschewski and Mulvey, within HHA Stone-Cech compactification can be defined for every locale.
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
