  We define and investigate spectral invariants for Floer homology $HF(H,U:M)$
of an open subset $U\subset M$ in $T^*M$, defined by Kasturirangan and Oh as a
direct limit of Floer homologies of approximations. We define a module
structure product on $HF(H,U:M)$ and prove the triangle inequality for
invariants with respect to this product. We also prove the continuity of these
invariants and compare them with spectral invariants for periodic orbits case
in $T^*M$.
