On certain separable and connected refinements of the Euclidean topology

Gerald Kuba

Write $c$ for the cardinality of the continuum and let $\eta$ be the Euclidean topology on ${\Bbb R}$. Let $\Sigma$ be the family of all $\sigma$-ideals ${\Cal I}$ on ${\Bbb R}$ such that $\bigcup{\Cal I}$ is dense and ${\Bbb Q}\cap\bigcup{\Cal I}=\emptyset$. Then for each ${\Cal I}\in\Sigma$ the family $\eta/{\Cal I}$ of all sets $X\setminus Y$ with $X\in\eta$ and $Y\in{\Cal I}$ is a topology on ${\Bbb R}$. Such a refinement of $\eta$ always preserves separability and connectedness, but destroys metrizability (and first countability almost always) and makes the space totally pathwise disconnected. Nevertheless, the separable Hausdorff space $({\Bbb R},\eta/{\Cal I})$ still has the two metric properties that every point is reachable by a sequence of points within any fixed countable dense set and that (even in the absence of first countability) sequential continuity is strong enough to entail continuity. In detail we investigate further main properties in the four most interesting cases when the $\sigma$-ideal ${\Cal I}$ consists of either all countable sets or all null sets or all meager sets or all sets contained in ${\Bbb R}\setminus{\Bbb Q}$. Finally we track down a subfamily $\Sigma_1$ of $\Sigma$ with cardinality $2^{2^c}$ such that $({\Bbb R},\eta/{\Cal I})$ and $({\Bbb R},\eta/{\Cal J})$ are never homeomorphic for distinct ${\Cal I},{\Cal J}$ in $\Sigma_1$.