A nowhere neighborhood nested
space is one in which no point has a local base which is linearly ordered by set
inclusion. An MI space is one in which every dense subset is open. In this paper we
show that every Hausdorff topology without isolated points has a nowhere
neighborhood nested refinement. We show that every maximal connected Hausdorff
topology is MI and nowhere neighborhood nested, and that every connected,
but not maximal connected, Hausdorff topology has a connected, but not
maximal connected, nowhere neighborhood nested refinement. Every connected
Hausdorff topology has a connected, MI, nowhere neighborhood nested
refinement.
