Vol. 57, No. 2, 1975

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Maximal connected Hausdorff spaces

J. Pelham Thomas

Vol. 57 (1975), No. 2, 581–583

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.

Mathematical Subject Classification 2000
Primary: 54D05
Received: 16 December 1974
Published: 1 April 1975
J. Pelham Thomas