 of outcomes that are possible when the property \(\Gamma\) obtains. There is a sense in which two properties are empirically indistinguishable if they have the same support: we cannot distinguish between them by means of a single execution of a single test. We might therefore wish to identify physical properties with classes of physically indistinguishable classical properties, or, equivalently, with their associated supports. However, if we wish to adhere to the programme of representing physical properties as subsets (rather than as equivalence-classes of subsets) of the state-space, we can do so, as follows. Define a mapping \(F : \wp(X) \rightarrow \wp(\Delta)\) by \(F(J) = \{\omega \in \Delta \mid S(\omega) \subseteq J \}\). The mapping \(\Gamma \rightarrow F(S(\Gamma))\) is then a closure operator on \(\wp(\Delta)\), and the collection of closed sets (that is, the range of \(F)\) is a complete lattice of sets, closed under arbitrary  intersection.[14]  Evidently, classical properties—subsets of \(\Delta\)—have the same support iff they have the same closure, so we may identify physical properties with closed subsets of the state-space: