On topological properties of families of finite sets

TL;DR

This paper investigates the topological structure of uniform families of finite sets, focusing on the Cantor-Bendixson index and homeomorphism conditions, revealing how these properties relate to the structure of the underlying set.

Contribution

It establishes new characterizations of when subfamilies are homeomorphic to the original family based on the presence of integer intervals, linking topological and combinatorial properties.

Findings

The Cantor-Bendixson index of certain subspaces is characterized.

Homeomorphism between families depends on containing intervals of integers.

Connections are made between topological properties and partition problems.

Abstract

We present results about the Cantor-Bendixson index of some subspaces of a uniform family F of finite subsets of natural numbers with respect to the lexicographic order topology. As a corollary of our results we get that for any omega-uniform family F the restriction F|M is homeomorphic to F iff M contains intervals of arbitrary length of consecutive integers. We show the connection of these results with a topological partition problem of uniform families.