Many coarse topologies on the real line

TL;DR

The paper explores a vast diversity of topologies on the real line, constructing many non-homeomorphic, Baire, and metrizable topologies within a specific family, and analyzes their lattice structures.

Contribution

It constructs numerous non-homeomorphic, completely normal, Baire, and metrizable topologies on the real line, revealing complex lattice and chain structures.

Findings

Constructed 2^c non-homeomorphic completely normal topologies

Among these, 2^c are Baire and 2^c are of first category

Constructed c non-homeomorphic completely metrizable topologies

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.