Miklos Schweitzer 1965_7
Source:
September 25, 2008
inductiongeometry3D geometryspherereal analysisreal analysis unsolved
Problem Statement
Prove that any uncountable subset of the Euclidean -space contains an countable subset with the property that the distances between different pairs of points are different (that is, for any points P_1 \not\equal{} P_2 and Q_1\not\equal{} Q_2 of this subset, \overline{P_1P_2}\equal{}\overline{Q_1Q_2} implies either P_1\equal{}Q_1 and P_2\equal{}Q_2, or P_1\equal{}Q_2 and P_2\equal{}Q_1). Show that a similar statement is not valid if the Euclidean -space is replaced with a (separable) Hilbert space.