On inductive limits of topological algebraic structures in relation to the product topologies (Q2751515)

Let \(X\) be an inductive limit space of an inductive system in a certain category \({\mathcal{C}}\) of topological spaces, of topological groups, of topological vector spaces, or of topological algebras, etc., endowed with the corresponding algebraic structure and with the inductive limit topology, denoted by \({\tau}^X_{\text{ind}}\). In this paper the authors study some properties concerning the harmonicity of the inductive limit topology \({\tau}^X_{\text{ind}}\) with the algebraic structure on \(X\). Furthermore, they consider an appropriate variant of \({\tau}^X_{\text{ind}}\) in each category \({\mathcal{C}}\) and study various kinds of harmonicity, and propose several problems. Certain properties of general topological spaces for the `commutativity' of (1) taking direct products and (2) taking inductive limits are also studied.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00042].