Obtainable sizes of topologies on finite sets

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.

• Type: Journal
• Year: 2010
• Where: JCT
• Authors: Kári Ragnarsson, Bridget Eileen Tenner
• Keywords: JCT 2010, Logarithmic Upper Bound, Smallest Neighborhood, Smallest Possible Number