On Tue, Feb 6, 2018 at 12:01 PM, George Janelidze <George.Janelidze@uct.ac.za> wrote:
Dear Colleagues,
Let me repeat from my exchange of massages with Steve Vickers:
As you know, a locale is called 0-dimensional if all its elements are joins of complemented ones. By a morphism L--->L' of locales I shall mean a map L'--->L that preserves all joins and finite meets (as usually). The inclusion functor
0-Dimensional locales--->Locales
has a left adjoint F, for which
F(L)={x in L | x is a join of complementary elements}.
Question: Is F semi-left-exact?
Can Example 1 in https://dml.cz/bitstream/handle/10338.dmlcz/119250/CommentatMathUnivCarolRet... be put to some use to answer the question negatively? It shows that the zero-dimensional reflection in topological spaces does not preserve finite products. The example uses fairly nice subspaces of R and R^2. With kind regards, Andrej [For admin and other information see: http://www.mta.ca/~cat-dist/ ]