Almost isometric embeddings into configuration-compacta (Q554384)

The author defines configuration-compactness of metric spaces. It is then shown that for a Polish metric space \(X\) this condition is sufficient for being AIE-free, i.e. given any almost isometric embedding from a metric space \(A\) to \(X\) there is actually an isometric embedding of \(A\) into \(X\). Moreover it is proved that if one assumes a continuous version of \(\aleph_0\)-homogeneity of the Polish metric space configuration-compactness is indeed equivalent to AIE-freeness, and is also sufficient for being almost isometry unique. Here a metric space \(X\) is called almost isometry unique if, whenever some metric space \(A\) is almost isometric to \(X\), the space \(A\) is actually isometric to \(X\).