categories: Re: Open maps of locales

["Dr. P.T. Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

On Tue, 6 Nov 2001, Walter Tholen wrote:

> I need help with respect to the following question.
> With the help of M.M. Clementino and J. Picado, I understood that an open
> morphism f: X --> Y in the category of locales (dual to the category of
> frames) has the property that taking inverse images (pullbacks) along
> f commutes with taking closures of sublocales:  f*[cl(N)] = cl(f*[N]) for
> all sublocales N of Y.
> Conversely, does this property force f to be open (as it does for
> topological spaces)? If not, is the answer positive when f is the
> embedding of a sublocale?
>
> I appreciate any suggestions/answers that you may have. Thanks,
>
> Walter Tholen.
>
The answer is no: for any locale Y, the embedding of the smallest dense
sublocale of Y has this property. Of course, if X is the smallest dense
sublocale of Y, then any sublocale of X is closed (since the frame
corresponding to X is Boolean); so this is tantamount to saying that,
given any sublocale N of Y, N and its closure have the same intersection
with X. This in turn is equivalent to the assertion that if we
have elements U and V of a frame satisfying V \leq U and
((U => 0) => 0) = U, then we also have ((U => V) => V) = U.
The verification of this is left as an exercise for the reader.

Peter Johnstone