On the word topology, and beyond

Today I (Jean G.-L.) have the pleasure to have a guest, Aliaume Lopez. We are going to talk about the word topology on X*. In the book, there is a so-called Topological Higman Lemma that says that, if X is a Noetherian space, then X* is Noetherian in the word topology, generalizing a famous theorem due to Graham Higman. However, there used to be no characterization of the word topology as a universal construction, say as a finest topology or a coarsest topology with some properties. Aliaume has managed to find a satisfactory, and simple, answer to this question. We will then discuss the case of infinite words, and we will end with a conjecture which, if true, would provide us with a large set of new Noetherian spaces. Read the full post.