2 research outputs found
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.Comment: Final version, to appear in Journal of Combinatorial Theory, Series