Ran space

whose underlying set is the set of all nonempty finite subsets of X: for a metric space X the topology is induced by the Hausdorff distance.

The notion is named after Ziv Ran.

In general, the topology of the Ran space is generated by sets for any disjoint open subsets

There is an analog of a Ran space for a scheme:[1] the Ran prestack of a quasi-projective scheme X over a field k, denoted by

, is the category whose objects are triples

consisting of a finitely generated k-algebra R, a nonempty set S and a map of sets

consist of a k-algebra homomorphism

and a surjective map

is a nonempty finite set of R-rational points of X "with labels" given by

A theorem of Beilinson and Drinfeld continues to hold:

A theorem of Beilinson and Drinfeld states that the Ran space of a connected manifold is weakly contractible.

[2] If F is a cosheaf on the Ran space

, then its space of global sections is called the topological chiral homology of M with coefficients in F. If A is, roughly, a family of commutative algebras parametrized by points in M, then there is a factorizable sheaf associated to A.

Via this construction, one also obtains the topological chiral homology with coefficients in A.

The construction is a generalization of Hochschild homology.