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Definition. A metric space is a set X together with a function d (called a metric or distance function) which assigns a real number d (x, y) to every pair x, y X satisfying the properties (or axioms): 1  d (x, y) 0 and d (x, y) = 0 x = y, 2  d (x, y) = d (y, x), 3  d (x, y) + d (y, z) d (x, z).
O'Meara [24] generalized the concept of âµ0âµ0-spaces as follows: A topological space X is called an âµ-space if it is regular and has a Ï-locally finite k-network. Any metrizable space is an âµ-space and all compact sets in âµ-spaces are metrizable. For further results, see [15]. The study of those locally convex spaces E which are weakly âµ-spaces (i.e. under the weak topology E has a Ï-locally finite k-network) is begun here for the important particular case of FrÃ©chet lcs.