This is a reference request, coming from someone with little knowledge of hyperfunctions:

What methods have been used to endow the space of hyperfunctions $\mathcal B(\mathbb R)$ with something like a topology?

Let me give some background to the question. In 1958, Sato famously introduced the space of hyperfunctions on $\mathbb R$ as a generalization of virtually all classical notions of functions (or distributions, or ...). His definition is very simple: It is the quotient space $$ \mathcal B(\mathbb R)=\mathcal O(\mathbb C\setminus \mathbb R)/\mathcal O(\mathbb C) $$ where $\mathcal O(U)$ denotes the space of holomorphic functions on some open subset $U\subset \mathbb C$. In other words, any hyperfunction is represented by a pair $(F^+,F^-)$ of holomorphic functions $F^+$ on the upper half-plane and $F^-$ on the lower half-plane; and two such pairs $(F^+,F^-)$ and $(G^+,G^-)$ define the same hyperfunction if $G^+-F^+$ and $G^- - F^-$ are the restriction of a single holomorphic function on all of $\mathbb C$. Naively, one may think of the hyperfunction defined by the pair $(F^+,F^-)$ as the ``boundary values of $F^+$ minus the boundary values of $F^-$''.

To see that this contains, at least, real-analytic functions, one notes that one can replace $\mathbb C$ above by any open subset $U\subset \mathbb C$ containing $\mathbb R$; then $$ \mathcal B(\mathbb R)\cong \mathcal O(U\setminus \mathbb R)/\mathcal O(U). $$ Now any real-analytic function on $\mathbb R$ is defined on some such $U$, and hence defines an element of $\mathcal O(U\setminus \mathbb R)$ by taking this extension of the real-analytic function on the upper half-plane, and $0$ on the lower half plane.

Any other function space carries a natural topology, but in the accounts I have looked at, the question of putting a topology on $\mathcal B(\mathbb R)$, or even just a notion of convergence, is avoided. For good reason: The inclusion $\mathcal O(\mathbb C)\hookrightarrow \mathcal O(\mathbb C\setminus \mathbb R)$ has dense image, so if we equip $\mathcal B(\mathbb R)$ with the quotient topology, it becomes indiscrete. On the other hand, I find it a bit surprising that this point is not discussed: After all, it is a function space, one wants to do functional analysis with it (like solving differential equations), so naturally one would like to talk about limits of (hyper)functions!

[Side remark, to indicate that this is a subtle point: A naive search for hyperfunctions will lead one to Graf's textbook "Introduction to hyperfunctions and their integral transforms". In this book, at the beginning of Chapter 2, Graf defines a notion of convergent sequence in $\mathcal B(\mathbb R)$, by declaring a sequence to converge in $\mathcal B(\mathbb R)$ if it can be lifted to a convergent sequence in $\mathcal O(\mathbb C\setminus \mathbb R)$. He leaves it as an exercise to the reader to show that limits are unique. Unfortunately, this is false -- as $\mathcal O(\mathbb C)\to \mathcal O(\mathbb C\setminus \mathbb R)$ has dense image by standard approximation results, any limit can be achieved.]

On the other hand, various subspaces of $\mathcal B(\mathbb R)$ have a nice topology:

1. For any closed subset $K\subset \mathbb R$, one can define the subspace of $K$-supported hyperfunctions $\mathcal B_K(\mathbb R)\subset \mathcal B(\mathbb R)$, as the kernel of a naturally defined restriction map $\mathcal B(\mathbb R)\to \mathcal B(\mathbb R\setminus K)$ (noting that the notion of hyperfunction makes sense also for open subsets of $\mathbb R$, and yields a sheaf). Then $\mathcal B_K(\mathbb R)$ has a natural topology as a nuclear Fréchet space. In particular, the subspace of compactly supported hyperfunctions has a natural topology as a sequential colimit of nuclear Fréchet spaces.

2. The space of distributions on $\mathbb R$ can be shown to embed naturally into $\mathcal B(\mathbb R)$. Again, the space of distributions has a nice topological structure.

The theory of condensed vector spaces handles this situation very nicely; like any quotient space, $\mathcal B(\mathbb R)$ has a natural structure as condensed $\mathbb R$-vector space and this makes the kernel of $\mathcal B(\mathbb R)\to \mathcal B(\mathbb R\setminus K)$ a condensed $\mathbb R$-vector space that comes (necessarily uniquely) from a nuclear Fréchet space.

In fact, that condensed mathematics can handle quotients by dense subspaces nicely is a thing I'm always stressing about the theory; but so far I wasn't aware of any such examples that are actually used in mathematical practice. Hyperfunctions do seem to be such a case!