Scattered and paracompact order topologies

TL;DR

This paper demonstrates that in ZFC, infinite sets can be endowed with a vast number of mutually non-homeomorphic scattered and compact order topologies, with the cardinality of such topologies being optimal.

Contribution

It establishes the existence of 2^|S| mutually non-homeomorphic scattered and compact order topologies on infinite sets within ZFC, including uncountable sets.

Findings

For infinite sets, 2^|S| complete metrics generate non-homeomorphic scattered order topologies.

Uncountable sets can have 2^|S| mutually non-homeomorphic scattered and compact order topologies.

The cardinality 2^|S| is shown to be optimal in both cases.

Abstract

We show that (in ZFC) every infinite set S can be equipped with 2^|S| complete metrics which generate mutually non-homeomorphic scattered order topologies on S. Furthermore, we show that (in ZFC) every uncountable set S can be equipped with 2^|S| mutually non-homeomorphic scattered and compact order topologies. (This would be unprovable in ZFC for countably infinite S.) In both enumeration theorems the cardinality 2^|S| is optimal.