Coarse topologies on the real line

Gerald Kuba

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 $\tau\in{\mathcal L}$ where $({\mathbb R},\tau)$ is {ı completely normal} resp.~{ı second countable} resp.~{ı not regular}. Trivially, ${\mathcal L}_1\cap{\mathcal L}_3=\emptyset$ and $|{\mathcal L}_i|\leq|{\mathcal L}|\leq 2^c$ and $|{\mathcal L}_2|\leq c$. For $\tau\in{\mathcal L}$ the space $({\mathbb R},\tau)$ is metrizable if and only if $\tau\in{\mathcal L}_1\cap{\mathcal L}_2$. 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-homeomorphic topologies $\tau\in{\mathcal L}_1$ the space $({\mathbb R},\tau)$ is {ı Baire}, but there are also $2^c$ non-homeomorphic topologies $\tau\in{\mathcal L}_1$ and $c$ non-homeomorphic topologies $\tau\in{\mathcal L}_1\cap{\mathcal L}_2$ where $({\mathbb R},\tau)$ is of {ı first category}. Furthermore, we investigate the {ı complete lattice} ${\mathcal L}_0$ of all topologies $\tau\in{\mathcal L}$ 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 topologies in ${\mathcal L}_0\cap{\mathcal L}_1\cap{\mathcal L}_2$ 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 $2^\kappa>c$.