Coarser connected topologies

Abstract

We investigate which spaces have coarser connected topologies. If in a collectionwise normal space X, the density equals the extent, which is attained and at least c , then X has a coarser connected collectionwise normal topology. In the previous sentence, the separation property collectionwise normal can be replaced by other separation properties—for example, Hausdorff, Urysohn, regular, metrizable. A zero-dimensional metrizable space X of density at least c has a coarser connected metrizable topology. A non-H-closed Hausdorff space with a σ-locally finite base has a coarser connected Hausdorff topology. We give necessary conditions and sufficient conditions for an ordinal to have a coarser connected Urysohn topology. In particular, every indecomposible ordinal of countable cofinality has a coarser connected topology. We present a nowhere locally compact Hausdorff space X with no coarser connected Hausdorff topology, yet X is dense in a connected Hausdorff space Y.