Many coarse topologies on the real line

Abstract

Let c denote the cardinality of the continuum. Let L denote the family of all Hausdorff topologies on the real line coarser than the natural topology. We construct 2^c pairwise non-homeomorphic completely normal topologies in L among which 2^c are Baire and 2^c are of first category. We also construct c pairwise non-homeomorphic completely metrizable topologies in L. Furthermore, we investigate complete lattices of topologies in L and construct extremely long chains of homeomorphic topologies in L.