On constructing topology from algebra

TL;DR

This thesis investigates methods to construct topological structures from semigroups, analyzing minimal and maximal topologies, and examining their uniqueness and automorphism groups.

Contribution

It introduces explicit descriptions of topologies on various semigroups and explores the conditions for unique Polish semigroup topologies.

Findings

Explicit descriptions of topologies on semigroups like monoids of relations and transformations

Criteria for the existence of unique Polish semigroup topologies

Proof of Rubin's theorem and automorphism group descriptions of Brin-Thompson groups

Abstract

In this thesis we explore natural procedures through which topological structure can be constructed from specific semigroups. We will do this in two ways: 1) we equip the semigroup object itself with a topological structure, and 2) we find a topological space for the semigroup to act on continuously. We discuss various minimum/maximum topologies which one can define on an arbitrary semigroup (given some topological restrictions). We give explicit descriptions of each these topologies for the monoids of binary relations, partial transformations, transformations, and partial bijections on a countable set. Using similar methods we determine whether or not each of these semigroups admits a unique Polish semigroup topology. We also do this for the various other semigroups, provide a proof of Rubin's theorem, and give a description of the automorphism groups of the Brin-Thompson groups. The thesis also contains many background results.