Structures induced from other structures

Question

As is well known, inner products induce norms, norms induce metrics and metrics induce topologies. I was wondering, are there any other well known (or common) mathematical structures that can be added to this chain (in either side)? $$ \text{Inner product} \longrightarrow \text{Norm} \longrightarrow \text{Metric} \longrightarrow \text{Topology} $$

Answer 1

You can insert "uniformity" between metric and topology, see wikipedia, e.g.. If you're more functional analysis minded, insert Fréchet spaces between normed spaces and metric spaces. They're an important class as well. There is an even weaker class, the metrisable linear spaces, inbetween Fréchet and metric (see topological vector spaces). the latter need not be complete nor locally convex, so are less useful in applications.

Also, unrelatedly to metrics or vector spaces, linear orders induce an order topology on a set.