Algebra Seminar talk
2023-05-26
Clemens Schindler
The semigroup of monotone functions on the rational numbers has a unique Polish topology
Abstract:
The space of all increasing functions on the rational numbers carries an algebraic structure (the standard composition operation of functions, yielding a semigroup) and a topological structure (the subspace topology inherited from the product topology on the power $\mathbb{Q}^\mathbb{Q}$ where each copy of $\mathbb{Q}$ carries the discrete topology, yielding a Polish topology). These structures are compatible in the sense that the operation is continuous with respect to the topology. Whenever one considers such a combined algebraic-topological structure, one can ask the following general question: Is the topology predetermined by the compatibility with the operations, i.e., does the algebra have a unique (Polish?) topology? In other words: Can the (Polish) topology be "reconstructed" from the algebra? I will present several examples of Polish groups and semigroups that do (not) have this property and sketch for some of them why this is the case. Afterwards, I will discuss why the successful techniques for these structures fail for the space of all increasing functions on the rational numbers and outline how this shortcoming can be mitigated to show that, indeed, the increasing functions on the rational numbers carry a unique Polish topology.
This is joint work with Michael Pinsker.