A topic I can write about is "Metric Models For Topology", based on
arxiv.org/abs/1311.4940 (to appear in Algebra Universalis). Basically, every topological space is metrizable if metrizable is interpreted as "metric space where the metric takes values in a value quantale rather than the non-negative real numbers". The result is a category of metric spaces which is equivalent to the category of topological spaces, thus providing a metric model for topology.