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 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 I 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 subcollection of the
collection of all 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
