Many coarse topologies on the real line
General Topology
2026-04-08 v1
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.
Cite
@article{arxiv.2604.05949,
title = {Many coarse topologies on the real line},
author = {Gerald Kuba},
journal= {arXiv preprint arXiv:2604.05949},
year = {2026}
}