S. D. V. Stephens

notes / Math 55b / chapter 3: Completeness, Separation and Quotients / From my notebook

N16 Separation Axioms

Definition N16.1 (Separation axioms).

  • is Hausdorff ( property) if for all points there are open sets , with .
  • is if is closed for every ; equivalently, for all there is an open with .
  • is regular () if it is and for all and all closed with , there are open , with .
  • is normal () if it is and for all closed with , there are open , with .
Four panels: T1 (an open set around y missing x), T2 (disjoint open sets around x and y), T3 (disjoint open sets around a point x and a closed set B), T4 (disjoint open sets around closed sets A and B).

Example N16.2. (the lower limit topology) is normal but not metrizable. is regular but not normal. (Munkres §31.)