Obtainable Sizes of Topologies on Finite Sets

TL;DR

This paper investigates the minimal size of topologies on finite sets with a given number of open sets, providing bounds and algorithms for constructing such topologies with specific properties.

Contribution

It introduces efficient algorithms to determine the smallest possible size of topologies with a given number of open sets and establishes bounds on these sizes.

Findings

The minimal size of a topology with k open sets has a logarithmic upper bound.

Existence of topologies on n points with k open sets for a wide range of k, exponentially large in n.

Algorithms can produce topologies with minimal neighborhood sizes for each point.

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.