Lattices in ℂ and Finite Subsets of a Circle

Abstract

2. SPACES OF FINITE SUBSETS. For a topological space X, let expk X be the set of all nonempty 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 that sends (xI, ..., xk) to {x } U ... U {xk }. The quotient topology gives expk X the structure of a topological space. Notice that when m < k the space expm X is canonically embedded in expk X. Clearly, exp, X = X. The simplest nontrivial example is provided by the space exp2 S1, which is homeomorphic to the Mdbius band. One way to see this is as follows [3]. Identify S' with the boundary of an open disk D in the projective plane. Notice that AM = RP2\D is a M~ibius band. For each point x of M there exist at most two lines that pass through x and are tangent to S1. Let T (x) in exp2 S1 be the corresponding set of tangency points. Then