The Third Symmetric Product of ℝ

Symmetric products of the line: Embeddings and retractions

“…In general X (k) is not a Lipschitz (or even topological) retract of X (n) . For example, if X is the circle S 1 , then X (3) is homeomorphic to S 3 [6] which, being simply connected, does not retract onto X (1) = S 1 . This suggests a potentially interesting problem.…”

“…In the context of metric spaces it is natural to seek embeddings that are bi-Lipschitz, not merely topological. Borovikova and Ibragimov proved in [3] that R (3) is lipeomorphic to R 3 ; previously these spaces were shown to be homeomorphic by Borsuk and Ulam [5]. Borovikova, Ibragimov and Yousefi [4] obtained partial results toward bi-Lipschitz embedding of R (n) into some Euclidean space R m .…”

Lipschitz clustering in metric spaces

“…where the elements of X(n) are subsets of X with at most n elements, and ∆ is the Hausdorff metric. The topology and geometry of X(n) can be difficult to grasp even for simple spaces X, e.g., [4,7,8,26]. This paper will show how the relations between the finite subset spaces X(n) reflect the cluster tendency of X.…”

Symmetric products of the Euclidean spaces and the spheres

“…Indeed, the map h : {(x, y) ∈ R 2 : x ≤ y} → F 2 (R) defined by h(x, y) = {x, y} is a homeomorphism. It was known that F 3 (R) and R 3 are homeomorphic, in particular, there is a bi-Lipschitz equivalence (see [6] or Section 4 for details). Turning toward the symmetric product F n (S 1 ) of the circle S 1 , in [10], it was proved that for n ∈ N, both F 2n−1 (S 1 ) and F 2n (S 1 ) have the 210…”