On constructing topology from algebra

TL;DR

The thesis develops a systematic framework for constructing and comparing topologies on algebraic objects—primarily semigroups and their clones—under compatibility constraints. It introduces five canonical topologies ($ ext{MF}$, $ ext{FM}$, $ ext{HM}$, $ ext{Z}$, $ ext{SCC}$) and analyzes their existence, uniqueness, and interactions across key semigroups, including binary relation monoids, full/partial transformation monoids, inverse/injective monoids, and various function/clones, with a focus on Polishness and completeness. Central results include the equivalence of comparable Polish topologies on groups, Rubin's theorem for reconstructing topological actions from algebra, and automatic continuity phenomena for large classes of maps, all extended to the clone setting. The work applies these tools to describe automorphism groups of Brin–Thompson groups and to connect topological dynamics with algebraic structure, yielding precise descriptions and invariants that can guide both theory and SEO-oriented search for topology-from-algebra results. Overall, it provides a coherent program for deriving canonical or unique topologies from algebraic data and demonstrates how such topologies illuminate the automorphism structure and continuity properties of rich algebraic systems.

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.

Figures (5)

• Figure 1: Part of a minimal transducer with $(X_2^*)^2$ as domain and range. This represents the baker's map in $2V$ (this transducer is infinite and every state has an edge with the core as its target).
• Figure 2: We define $f_\phi:= f_{T, q_0}$ where $T$ is the left transducer, and $f_{\phi^{-1}}:= f_{T, q_0}$ where $T$ is the right transducer
• Figure 3: Two transducers with domain and range $X_2^*$
• Figure 4: The transducer obtained by taking the categorical product of the transducers in Figure \ref{['figure1']}
• Figure 5: A single state $(1, 4, 2, 2)$-transducer defining a homeomorphism from $\mathfrak{C}_{4}$ to $\mathfrak{C}_2^2$ (analogous to those in Figure \ref{['figure7']})

Theorems & Definitions (376)

• Definition 1.1: Numbers
• Definition 1.2: Power set
• Definition 1.3: Functions, Assumed Knowledge: \ref{['power set defn']}
• Definition 1.4: Function sets, Assumed Knowledge: \ref{['function defn']}
• Definition 1.5: Product sets and projection maps, Assumed Knowledge: \ref{['function sets def']}
• Definition 1.6: Binary relations, Assumed Knowledge: \ref{['power set defn']}
• Definition 1.7: Countable, Assumed Knowledge: \ref{['Numbers defns']}, \ref{['binary relations defns']}
• Definition 1.8: Union and intersection conventions
• Definition 1.9: Complement
• Definition 1.10: Types of binary relation, Assumed Knowledge: \ref{['binary relations defns']}