## Covering Properties of Quasi-Uniform Spaces

If (X,T) is a compact metrizable space, then every metric that induces T has the following property. For each open cover C of X there is a positive real number λ(C), called a Lebesgue number of C, such that if A ⊂ X and diam (A) ≤ λ(C), then there exists C ε C such that A ⊂ C. In general, if (X,T) is a metrizable space that is not compact, an open cover C may have a Lebesgue number for one metric and fail to have a Lebesgue number for some other metric. What is required of a metric in order that every open cover have a Lebesgue number is that for each open cover C there is an entourage U of the induced metric uniformity such that {U(x)|x ε X} refines C. This property of metric uniformities is important for quasi-uniformities as well.