Recognition and reconstruction of sets in ^2 via their projections - Barov, Stoyu (Topology Proceedings, 2024)
Let k N and let P be a subset of all k-dimensional linear subspaces G_k of ^2 with the natural topology. The subsets B and C of ^2 are called P-imitations of each other if B+P=C+P for every P P. In the case when P is somewhere dense G_δ-set in G_k, we show that there are certain non-trivial sets in ^2 such that each of them has only one P-imitation, namely, itself. Consequently, every such set can be reconstructed as the intersection of the preimages of its projections under P. In addition, we discuss some important properties of σ-compact sets that are of independent interest.