Obtainable Sizes of Topologies on Finite Sets
Combinatorics
2009-05-20 v4 General Topology
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.
Keywords
Cite
@article{arxiv.0802.2550,
title = {Obtainable Sizes of Topologies on Finite Sets},
author = {Kari Ragnarsson and Bridget Eileen Tenner},
journal= {arXiv preprint arXiv:0802.2550},
year = {2009}
}
Comments
Final version, to appear in Journal of Combinatorial Theory, Series A