  We extend the scope of noncommutative geometry by generalizing the
construction of the noncommutative algebra of a quotient space to situations in
which one is no longer dealing with an equivalence relation. For these
so-called tolerance relations, passing to the associated equivalence relation
looses crucial information as is clear from the examples such as coarse
graining in physics or the relation $d(x,y)< \varepsilon$ on a metric space.
Fortunately, thanks to the formalism of operator systems such an extension is
possible and provides new invariants, such as the $C^*$-envelope and the
propagation number.
  After a thorough investigation of the structure of the (non-unital) operator
systems associated to tolerance relations, we analyze the corresponding state
spaces. In particular, we determine the pure state space associated to the
operator system for the relation $d(x,y)< \varepsilon$ on a path metric measure
space.
