A NOTE ON LEBESGUE SPACES

L~t (X,d) be a metric space. A Lebesgue number for an open cover lj of X is an E> 0 such that for each point p E X, the open,ball Bd(p,E) = {x E X: d(p,x) < E} is con tained in at least one member of 0. A Lebesgue space (L-space)(is a metric space such that every open cover of the space has a Lebesgue number. It is known that the L-spaces are precisely those metric spaces for which every continuous real-valued function is uniformly continuous ([6, p. 112], [1, p. 12]). In particular, every compact metric space is an L-space. We will show that there are L-spaces which are not even locally compact. Furthermore, in Theorem 1 we characterize those metric spaces which con tain nonlocally compact L-subspaces. The proof of Theorem 1 shows how to construct such subspaces in the most general possible setting. In Theorem 2 we show that any L-spac