An approach to harmonic analysis on non-locally compact groups I: level structures over locally compact groups

We define a class of spaces on which one may generalise the notion of compactness following motivating examples from higher-dimensional number theory. We establish analogues of several well-known topological results (such as Tychonoff's Theorem) for such spaces. We also discuss several possible applications of this framework, including the theory of harmonic analysis on non-locally compact groups.