What is meant by an open boundary when specifying boundary conditions of PDEs? When speaking about boundary conditions of  PDEs, one speaks about Dirichlet, Neumann or Cauchy boundary conditions specified over the boundary which can be closed or open. 
For example,  we say that for hyperbolic partial differential equations, specification of Cauchy boundary condition on an open boundary gives rise to unique, stable solutions.
What is meant by an open boundary here? How can a boundary of the region on which we solve our PDE  be open? Surely the region is a specified region of space and obviously has a closed boundary. (part of which may be at infinity of course) 
 A: I'm afraid that the name "open boundary" is not very fortunate for someone with  internalized topological language. Page 13 of the book Mathematical Methods in Physics and Engineering with Mathematica talks about boundary curves or surfaces. It states that

"An open boundary is one that doesn't completely enclose the domain
  but lets it extend to infinity in at least one direction",

whereas a closed boundary is one that "surrounds the domain everywhere, confining it to a finite surface or volume". It remarks that the Cauchy condition actually represents an initial condition, which makes sense if you think of the unbounded direction as time. 
I would guess that these names predate the topological "open" and "closed" notions, because they look like classical usage to me. Maybe someone else can clarify this point.
A: Consider a boombox. Turn on the boombox. If you want to model the sound waves coming out of the boombox you'd consider solving the wave equation over $\mathbb R^2$ or $\mathbb R^3$. Those domains are open.
Perhaps you're thinking about it too physically. Mathematically it's totally valid to have an infinite string or an infinite heating rod. Whether it has physical meaning or not is up to the physicist to decide(but surprisingly most of the time it does!)
P.S. OP is clearly aware that boundaries can tend to infinity, so I'm not quite sure where the  confusion lies.
