Local decomposability is a sort of separation axiom, like a weak sort of regularity, that is trivial in classical mathematics, but interesting in constructive mathematics.
A topological space is locally decomposable if for any open set and point , there exists an open set with such that for all we have either or . If excluded middle holds, of course, we can take .
For point-set apartness spaces, which are equivalent to certain topological spaces, the condition can be rephrased as: if , then there is a set such that and for all we have either or .
For uniform spaces, the notion of uniform regularity is really a notion of “uniform local decomposability”; but since in the uniform case it is sufficient to imply full regularity, we generally call it “uniform regularity” instead. For quasi-uniform spaces this is no longer true (since after all, there are non-regular quasi-uniform spaces classically), so we should speak of uniform local decomposability instead.
Last revised on January 18, 2017 at 09:23:26. See the history of this page for a list of all contributions to it.