Continuous Truth I Non-constructive Objects

174 M.P. FOURMAN
that is
so
W x U(k3V E Kln2.rl E V.
IK" = {Wi x Uilfor some Vi c U(W) we have Wi x Ui It Vi E K1r2 A rl c Vi}
covers W x U. [Because, if p: X + Y is an open surjection and Xlk VEKJPA elp E V
then YlkV E K A e E V: the basic opens of )# are constant and thus descend open
surjections.1 But now we claim IK* IK because, by definition
wi x ui It n1 E vi iff Mi 5 Vi
and, as projections are covers,
Wi x Uilb Vi E KIn2 iff UiIkVi E KIUi. 0
Special cases of this are worthy of mention. When X is Baire space NN, Cantor
N
space 2 , Dedekind reals R, to say that X has enough points is the internal
statement of Bar induction, Fan theorem, Heine Bore1 theorem (respectively). For
these cases it is sufficeint to take the topology on Q generated by covering
families of open inclusions: since each of these spaces X has a point the proections
X x U + X are covers for this topology. We call this topology the open
inclusion topology.
We introduce some more general spaces. Let f: X + U in LOC. We consider the
internal locale X/f defined at U by the basis U(V) with all its standard covers.
More properly for 9: W + U we define
W/f)lg = 0(9*X)
given by pulling back f along g. Any commuting triangle
x
F Y
induces an internal map of locales
5: X/f + Y/h
defined at U. Given by 5-l on basis elements, this clearly takes basic covers to
covers. Furthermore, if 5: X + Y is open (and surjective) then 5: X/f + Y/h is
open, since it suffices to define 4 comnuting with A on basis elements, (and
surjective since if 5: X + Y is an open surjection then so are all its pullbacks,
so internally 5-l reflects basic open covers). These spaces include the spaces )K
we introduced earlier as
U1l-X (X x U)/v .
We now specialise to the case where the objects of b are TI.
an isomorphism
(ptX)(U) CCUj,Xl CCU,Xl
Then U 4 U j induces
so that X represents the functor ptX. This happens in particular for the spaces
N, NN, pN, R and their basic opens (see Fourman 119831.) Furthermore, any element
of pt(X/f) defined at U induces a comnuting triangle