A uniform space is trans-separable if every uniform cover has a countable subcover. We show that a uniform space is trans-separable if it contains a suitable family of precompact sets. Applying this result to locally convex spaces, we are able to deduce that the precompact subsets of a wide class of spaces are metrisable. The proof of our main Theorem is based on a cardinality argument, and is reminiscent of the classical Bolzano-Weierstrass Theorem.A uniform space X is said to be trans-separable if every uniform cover of X admits a countable subcover. Expressed in terms of entourages or surroundings, this means that for each entourage H in X x X, there is a countable subset C of X satisfying H[C\ = X. This condition is strictly weaker than the usual topological notion of separability: every separable uniform space is trans-separable, but not conversely. The class of transseparable spaces forms a reflective subcategory of the category of uniform spaces; it is closed under the taking of completions, subspaces, products, uniformly continuous images and projective limits. Every Lindelof uniform space is trans-separable, but again the converse does not hold.The concept of trans-separability is sometimes encountered in the context of topological vector spaces. Pfister [6] has shown that a locally convex space is trans-separable if and only if the equicontinuous subsets of its dual are weak* metrisable. (For simplicity, we shall assume that all our spaces are Hausdorff.) As a consequence, many results in functional analysis which are normally stated in terms of separable spaces actually only require the weaker condition of trans-separability.Trans-separability appears under many different names in the literature. While workers in the field of uniform spaces tend to speak simply of separable spaces (see [3] and [4]), those who study locally convex spaces favour the phrase "separable by seminorm". Pfister investigated topological vector spaces which were "of countable type", whereas Drewnowski [2] coined the word "trans-separable" while working with topological abelian groups.Pfister used the idea of trans-separability to show that in a DF space precompact sets are metrisable. (For the definition of a DF space, and other terms from the theory

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By the term "locally convex space", we mean a locally convex Hausdorff topological vector space (see [17]). We shall denote the algebraic dual of a locally convex space E by E*, and its topological dual by E'. It is convenient to think of the elements of E as being linear functionals on E', so that E can be identified with a subspace of E'*.