© 1996 Copyright by Arthur Stone. All rights reserved.

Question

Finite Closed Covers of the Sphere of Unit Radius in Rn Arthur Stone

J. Tiser asked orally in 1991:
Given a metric space (X,\rho) and a decreasing sequence a₁ >= a₂ >= a₃ ... of positive real numbers converging to zero, does there exist a sigma-discrete open cover of X say U= \cup {U_n: n \in N} such that, for each n, U_n is discrete and each U \in U_n has diameter less than a_n ?

Stone proved yes if dimX < \infty in his preprint entitled ``Covering Dimension From Large Sets''.

Stone also showed that every finite closed cover of the unit cube in Rⁿ by sets of diameter less than 1 has order at least n. He asks if ``less than'' can be replaced by ``at most''.

Stone also showed that every finite closed cover of the sphere of unit radius in Rⁿ by sets of diameter less than c has order at least n when c = 2/\surdn. Stone conjectures this remains true when c = 2.