Separation axioms and frame representation of some topological facts (Q1330918)

It is well known that the ``algebraic'' (i.e. frame-theoretic) versions of certain familiar properties of topological spaces and continuous maps are not quite equivalent to the traditional topological ones, but become so under the assumption that the spaces involved satisfy mild separation properties such as sobriety or the \(T_ D\) axiom of Aull and Thron. Here the authors observe that, in a number of cases, these separation axioms (or their ``non-\(T_ 0\)'' versions) are necessary as well as sufficient for the equivalences to hold.