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 the set N. Using similar methods we determine whether or not each of these semigroups admits a unique Polish semigroup topology. We also do this for the following semigroups: the monoid of all injective functions on N, the monoid of continuous transformations of the Hilbert cube [0, 1]N, the monoid of continuous transformations of the Cantor space 2N, and the monoid of endomorphisms of the countably infinite atomless boolean algebra. With the exception of the continuous transformation monoid of the Hilbert cube, we also show that all of the above semigroups admit a second countable semigroup topology such that every semigroup homomorphism from the semigroup to a second countable topological semigroup is continuous. In a recent paper, Bleak, Cameron, Maissel, Navas, and Olukoya use a theorem of Rubin to describe the automorphism groups of the Higman-Thompson groups Gโ,แตฃ via their canonical Rubin action on the Cantor space. In particular they embed these automorphism groups into the rational group R of transducers introduced by Grigorchuk, Nekrashevich, and Sushchanskii. We generalise these transducers to be more suitable to higher dimensional Cantor spaces and give a similar description of the automorphism groups of the Brin-Thompson groups ๐Vโ (although we do not give an embedding into R). Using our description, we show that the outer automorphism group Out(๐Vโ) of ๐Vโ is isomorphic to the wreath product of Out(1Vโ) with the symmetric group on ๐ points.

On constructing topology from algebra

Elliott, L. D. B. (Author). 14 Jun 2022

Student thesis: Doctoral Thesis (PhD)