### Metric Geometry of Finite Subset Spaces [article]

Earnest Akofor
2021 arXiv   pre-print
If X is a (topological) space, the nth finite subset space of X, denoted by X(n), consists of n-point subsets of X (i.e., nonempty subsets of cardinality at most n) with the quotient topology induced by the unordering map q:X^n→ X(n), (x_1,⋯,x_n)↦{x_1,⋯,x_n}. That is, a set A⊂ X(n) is open if and only if its preimage q^-1(A) is open in the product space X^n. Given a space X, let H(X) denote all homeomorphisms of X. For any class of homeomorphisms C⊂ H(X), the C-geometry of X refers to the
more » ... ption of X up to homeomorphisms in C. Therefore, the topology of X is the H(X)-geometry of X. By a (C-) geometric property of X we will mean a property of X that is preserved by homeomorphisms of X (in C). Metric geometry of a space X refers to the study of geometry of X in terms of notions of metrics (e.g., distance, or length of a path, between points) on X. In such a study, we call a space X metrizable if X is homeomorphic to a metric space. Naturally, X(n) always inherits some aspect of every geometric property of X or X^n. Thus, the geometry of X(n) is in general richer than that of X or X^n. For example, it is known that if X is an orientable manifold, then (unlike X^n) X(n) for n>1 can be an orientable manifold, a non-orientable manifold, or a non-manifold. In studying geometry of X(n), a central research question is "If X has geometric property P, does it follow that X(n) also has property P?". A related question is "If X and Y have a geometric relation R, does it follow that X(n) and Y(n) also have the relation R?". (Truncated)