There are plenty of well-known, well-behaved topologies on the space $\mathcal{B}(H)$ of bounded operators on a Hilbert space. I am wondering whether there are any topologies on the space of (possibly) unbounded operators on $H$ which people typically use.
The issue is that unbounded operators are defined as partial functions, so one has to constantly worry about the domain on which they are defined. My idea was to use the correspondence that partial functions $f:X\to Y$ are in bijection with functions $f:X\to Y\cup \{\star\}$, where domain$(f)=X\setminus f^{-1}(\star)$. In other words, elements $x\in X$ for which $f(x)$ is not defined in $Y$ are defined to be $f(x)=\star$.
In the case that $Y$ has a (Hausdorff, non-compact) topology, one can define the one point compactification topology on $Y\cup\{\star\}$. Can we view an unbounded operator $A:H\to H$ as a continuous function $H\to H\cup\{\star\}$ into the one point compactification of $H$? If so, we could put the compact-open topology on $C(H,H\cup\{\star\})$ and then endow the set of unbounded linear operators with the subspace topology. Does this lead to a reasonably well-behaved topology?
There are potentially other ways to compactify $H$ too, such as the radial compactification. I can't seem to find much about topologizing the space of unbounded operators, so any help would be appreciated.
You may be interested in the result that the collection of all closed linear operators $\mathsf C(\mathcal H)$ forms a topological space as it can be equipped with a metric. For this refer to Chapter IV.2.4 in Kato's book "Perturbation Theory for Linear Operators" (1980) and the references therein:
One can show that the map \begin{align*} \Delta:\mathsf C(\mathcal H)\times\mathsf C(\mathcal H)&\to[0,\infty)\\(S,T)& \mapsto \delta({\rm gr}(S),{\rm gr}(T)) \end{align*} (where \begin{align*} \delta(M,N):=\max\big\{ \sup_{u\in M,\|u\|=1}\inf_{v\in N}\|u-v\| , \sup_{v\in N,\|v\|=1}\inf_{u\in M}\|u-v\|\big\} \end{align*} for any closed subspaces $M,N$ of $\mathcal H\times\mathcal H$) is a metric on $\mathsf C(\mathcal H)$, cf. footnote 1 on page 198 of Kato's book. Note that $\delta$ is almost the Hausdorff distance, the only difference being that the supremum is restricted from the full space to the unit sphere. Finally it is worth mentioning that $\mathsf C(\mathcal H)$ is not complete with respect to $\Delta$ (page 202, Remark 2.10).