  We develop a discrete Morse theory for open simplicial complexes
$K=X\setminus T$ where $X$ is a simplicial complex and $T$ a subcomplex of $X$.
A discrete Morse function $f$ on $K$ gives rise to a discrete Morse function on
the order complex $S_K$ of $K$, and the topology change determined by $f$ on
$K$ can be understood by analyzing the the topology change determined by the
discrete Morse function on $S_K$. This topology change is given by a structure
theorem on the level subcomplexes of $S_K$. Finally, we show that the
Borel-Moore homology of $K$, a homology theory for locally compact spaces, is
isomorphic to the homology induced by a gradient vector field on $K$ and deduce
corresponding weak Morse inequalities.
