Basic Intervals in the Partial Order of Metrizable Topologies

For a set X, let Σm(X) denote the set of metrizable topologies on X, partially ordered by inclusion. We investigate the nature of intervals in this partial order, with particular emphasis on basic intervals (in other words, intervals in which the topology changes at at most one point). We show that there are no non-trivial finite intervals in Σm(X) (indeed, every such interval contains a copy of P(ω)/fin). We show that although not all intervals in Σm(X) are lattices, all basic intervals in Σm are lattices. In the case where X is countable, we show that there are at least two isomorphism classes of basic intervals in Σm(X), and assuming the Continuum Hypothesis there are exactly two such isomorphism classes.