Tue, 28 Nov 2023, 1:25 pm MT

The set 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 topology of pointwise convergence in which a sequence of functions converges if and only if it is eventually constant at every argument). 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 introduce the known proof techniques. Afterwards, I will discuss why these successful techniques 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.