Weaker connected and weaker nowhere locally compact topologies for metrizable and similar spaces (Q1612183)

The authors extend the result that every noncompact second countable regular space has a weaker Hausdorff connected topology. To this end they first prove that any metrizable noncompact space has a weaker metrizable nowhere locally compact topology. As a consequence they conclude that any metrizable noncompact space has a weaker Hausdorff connected topology. Finally they establish that the same holds for any Hausdorff space \(X\) with a \(\sigma\)-locally finite base whose weight \(w(X)\) is a successor cardinal. Many interesting questions related to these results are asked.