Lattices in ℂ and Finite Subsets of a Circle

For X a topological space let expk X be the set of all non-empty finite subsets of X of cardinality at most k. There is a map from the Cartesian product of k copies of X with itself to expk X which sends x1 xk to x1 xk . The quotient topology gives expk X the structure of a topological space. Notice that for any m k the space expmX is canonically embedded into expk X . Clearly, exp1X X . The simplest non-trivial example is provided by the space exp2 S1 which is homeomorphic to the Möbius band. One way to see it is as follows ([3]). Let us identify S1 with the boundary of an open disk D in the projective plane. Notice thatM RP2 D is a Möbius band. For each point x M there exist at most two lines that pass through x and have a tangency with S1. Let T x exp2 S1 be the corresponding set of tangency points. Then T :M exp2 S1 is the desired homeomorphism. Notice that the boundary of the band corresponds to one-point subsets. The space exp3 S1 is described by a theorem of R. Bott: