  We give a permutation model in which Stone's Theorem (every metric space is
paracompact) is false and the Boolean Prime Ideal Theorem (every ideal in a
Boolean algebra extends to a prime ideal) is true. The erring metric space in
our model attains only rational distances and is not metacompact. Transfer
theorems give the comparable independence in the Zermelo-Fraenkel setting,
answering a question of Good, Tree and Watson.
