In mathematics, different variants of weak topologies on certain vector spaces or their subsets are ubiquitous (e.g. the weak$^*$ topology on the convex set of probability measures $\mathcal{P}(X)$ of some compact space $X$). However, by the nature of these topolgies, they are locally convex (they are defined by seminorms).
There are other natural examples of locally convex topologies defined on common function spaces (say $C^\infty(\mathbb{R})$ with convergence on compact sets) (or $\mathcal{S}(R)$ the space of Schwartz functions with certain norms defined by comparison to polynomials). However, I don't know of many non locally convex topolgies, one of the notable being $L_p$ for $0<p<1$.
my question is Are there examples of important non locally convex topologies used in mathematics?