  We introduce the notion of a pre-spectral triple, which is a generalisation
of a spectral triple $(\mathcal{A}, H, D)$ where $D$ is no longer required to
be self-adjoint, but closed and symmetric. Despite having weaker assumptions,
pre-spectral triples allow us to introduce noncompact noncommutative geometry
with boundary. In particular, we derive the Hochschild character theorem in
this setting. We give a detailed study of Dirac operators with Dirichlet
boundary conditions on open subsets of $\mathbb{R}^d$, $d \geq 2$.
