Let (X,p) be a compact metric space. Let C be a collection of open subsets of X such that C covers X, i.e., X= Union of Q where Q belongs to C. Prove that there exists a strictly positive real number r, such that for each point a in X, there is a member Q of the collection C such that an open ball Br(a) of radius r centered at 'a' is contained in Q, i.e. Br(a) is a strict subset of Q. Any ideas how to do this question? Thanks!