There exist 2^c pairwise non-homeomorphic completely normal topologies on the reals coarser than the standard topology, with 2^c Baire and 2^c first category, plus c completely metrizable ones, along with results on lattices and chains.
arXiv:2006.02880v1
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Many coarse topologies on the real line
There exist 2^c pairwise non-homeomorphic completely normal topologies on the reals coarser than the standard topology, with 2^c Baire and 2^c first category, plus c completely metrizable ones, along with results on lattices and chains.