Coarse topologies on the real line

Abstract

Let $c=|\mathbb{R}|$ denote the cardinality of the continuum and let $\eta$ denote the Euclidean topology on $\mathbb{R}$.Let $\mathcal{L}$ denote the family of all Hausdorff topologies $\tau$ on $\mathbb{R}$ with $\tau \subset \eta$.Let ${\mathcal{L}}_1$ resp.~${\mathcal{L}}_2$ resp.~${\mathcal{L}}_3$ denote the family of all $\tau ın\mathcal{L}$ where $(\mathbb{R},\tau )$ is{ıt completely normal} resp.~{ıt second countable} resp.~{ıt not regular}. Trivially, ${\mathcal{L}}_1\cap {\mathcal{L}}_3=\mathrm{\varnothing }$ and$|{\mathcal{L}}_i|\le |\mathcal{L}|\le 2^c$ and $|{\mathcal{L}}_2|\le c$. For $\tau ın\mathcal{L}$ the space $(\mathbb{R},\tau )$ is metrizableif and only if $\tau ın{\mathcal{L}}_1\cap {\mathcal{L}}_2$. We show that, up to homeomorphism, both ${\mathcal{L}}_1$ and ${\mathcal{L}}_3$ contain precisely $2^c$topologies and ${\mathcal{L}}_2$ contains precisely $c$ completely metrizable topologies. For $2^c$ non-homeomorphictopologies $\tau ın{\mathcal{L}}_1$ the space $(\mathbb{R},\tau )$ is {ıt Baire}, but there are also $2^c$ non-homeomorphic topologies$\tau ın{\mathcal{L}}_1$ and $c$ non-homeomorphic topologies $\tau ın{\mathcal{L}}_1\cap {\mathcal{L}}_2$ where $(\mathbb{R},\tau )$ is of {ıt first category}.Furthermore, we investigate the {ıt complete lattice} ${\mathcal{L}}_0$ of all topologies $\tau ın\mathcal{L}$ such that $\tau$ and $\eta$coincide on $\mathbb{R}\setminus \{0\}$. In the lattice ${\mathcal{L}}_0$ we find $2^c$ (non-homeomorphic) immediate predecessors of the maximum $\eta$,whereas the minimum of ${\mathcal{L}}_0$ is a compact topology without immediate successors in ${\mathcal{L}}_0$. We construct chains of homeomorphic topologiesin ${\mathcal{L}}_0\cap {\mathcal{L}}_1\cap {\mathcal{L}}_2$ and in ${\mathcal{L}}_0\cap {\mathcal{L}}_2\cap {\mathcal{L}}_3$ and in ${\mathcal{L}}_0\cap ({\mathcal{L}}_1\setminus {\mathcal{L}}_2)$ and in ${\mathcal{L}}_0\cap ({\mathcal{L}}_3\setminus {\mathcal{L}}_2)$ such that the length of each chain is $c$ (and hence maximal). We also track down a chain in ${\mathcal{L}}_0$ of length $2^{\lambda }$ where $\lambda$ is the smallest cardinal number $\kappa$ with $2^{\kappa }>c$.