[Paper Review] Topological Transformation Monoids

Zachary Mesyan, James D. Mitchell+1 more|arXiv (Cornell University)|Sep 12, 2018

Advanced Topology and Set Theory15 references3 citations

TL;DR

This paper investigates semigroup topologies on the full transformation monoid $\Omega^\Omega$ of an infinite set $\Omega$, establishing that the pointwise topology is the weakest Hausdorff semigroup topology and the unique Polish semigroup topology when $\Omega$ is countable. It further shows that no locally compact perfect Hausdorff semigroup topology exists on $\Omega^\Omega$ if $|\Omega|$ has uncountable cofinality.

ABSTRACT

We investigate semigroup topologies on the full transformation monoid T(X) of an infinite set X. We show that the standard pointwise topology is the weakest Hausdorff semigroup topology on T(X), show that the pointwise topology is the unique Hausdorff semigroup topology on T(X) that induces the pointwise topology on the group of all permutations of X, and construct |X| distinct Hausdorff semigroup topologies on T(X). In the case where X is countable, we prove that the pointwise topology is the only Polish semigroup topology on T(X). We also show that every separable semigroup topology on T(X) is perfect, describe the compact sets in an arbitrary Hausdorff semigroup topology on T(X), and show that there are no locally compact perfect Hausdorff semigroup topologies on T(X) when |X| has uncountable cofinality.

Motivation & Objective

• To determine the weakest Hausdorff semigroup topology on the full transformation monoid $\Omega^\Omega$ for an infinite set $\Omega$.
• To characterize the uniqueness of the pointwise topology as the only semigroup topology inducing the pointwise topology on the symmetric group $\mathrm{Sym}(\Omega)$.
• To investigate the existence and structure of Polish, separable, and locally compact semigroup topologies on $\Omega^\Omega$.
• To analyze compact sets and topological properties such as perfectness and cofinality constraints in Hausdorff semigroup topologies.

Proposed method

• Establishing that the pointwise topology is the weakest $T_1$ (and hence Hausdorff) semigroup topology on $\Omega^\Omega$ via a generalization of Gaughan’s result for symmetric groups.
• Using the fact that any $T_1$ semigroup topology on $\Omega^\Omega$ must contain the pointwise topology, enabling comparison with known topological properties.
• Applying the Cantor-Bendixson theorem and cofinality arguments to analyze compactness and local compactness in $T_1$ semigroup topologies.
• Proving that in a perfect $T_1$ topology, the image sets $\Sigma_\alpha = (\alpha)X$ for compact $X$ must be finite, leading to contradictions under uncountable cofinality.
• Constructing $|\Omega|$ distinct Hausdorff semigroup topologies on $\Omega^\Omega$ via specific topological constructions.
• Using the structure of semitopological semigroups and properties of nowhere dense sets to derive constraints on compact sets and bases of topologies.

Experimental results

Research questions

• RQ1 Is the pointwise topology the weakest Hausdorff semigroup topology on $\Omega^\Omega$?
• RQ2 Is the pointwise topology the unique Hausdorff semigroup topology on $\Omega^\Omega$ that induces the pointwise topology on $\mathrm{Sym}(\Omega)$?
• RQ3 Does $\Omega^\Omega$ admit a Polish semigroup topology other than the pointwise topology when $\Omega$ is countable?
• RQ4 Can there exist a locally compact perfect Hausdorff semigroup topology on $\Omega^\Omega$ when $|\Omega|$ has uncountable cofinality?
• RQ5 What are the structural constraints on compact sets in arbitrary Hausdorff semigroup topologies on $\Omega^\Omega$?

Key findings

• The pointwise topology is the weakest $T_1$ (and hence Hausdorff) semigroup topology on $\Omega^\Omega$, and this holds for any subsemigroup containing elements of rank 1 and 2.
• The pointwise topology is the unique $T_1$ semigroup topology on $\Omega^\Omega$ that induces the pointwise topology on $\mathrm{Sym}(\Omega)$, generalizing Gaughan’s result for symmetric groups.
• When $\Omega$ is countable, the pointwise topology is the only Polish semigroup topology on $\Omega^\Omega$, implying uniqueness of the Polish structure.
• Every separable semigroup topology on $\Omega^\Omega$ is perfect, meaning that the closure of any nonempty open set has nonempty interior.
• In any Hausdorff semigroup topology on $\Omega^\Omega$, compact sets are closed and satisfy $|\alpha X| < \aleph_0$ for all $\alpha \in \Omega$, meaning each point has finite image under the action of the set.
• There are no locally compact perfect Hausdorff semigroup topologies on $\Omega^\Omega$ when $|\Omega|$ has uncountable cofinality, as shown by a contradiction involving image sets and cofinality constraints.