Obtainable sizes of topologies on finite sets

  • Authors:

  • Kári Ragnarsson;Bridget Eileen Tenner

  • Affiliations:

  • Mathematics Institute, Reykjavík University, Kringlunni 1, 103 Reykjavík, Iceland;Department of Mathematical Sciences, DePaul University, 2320 North Kenmore Avenue, Chicago, IL 60614, USA

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2010

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Abstract

We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a topology with a prescribed size, we show that this number has a logarithmic upper bound. We deduce that there exists a topology on n points having k open sets, for all k in an interval which is exponentially large in n. The construction algorithms can be modified to produce topologies where the smallest neighborhood of each point has a minimal size, and we give a range of obtainable sizes for such topologies.