Given a metrizable space X of density κ, we study the topological structure of the space PM(X) of continuous bounded pseudometrics on X, which is endowed with the topology of uniform convergence. We prove that PM(X) is homeomorphic to [0,1)κ(κ−1)/2 if X is finite, to ℓ2(2<κ) if X is infinite and generalized compact, and to ℓ2(2κ) if X is not generalized compact. We also show that for an infinite σ-compact metrizable space X, the space M(X)⊂PM(X) of continuous bounded metrics on X and the space AM(X)⊂M(X) of bounded admissible metrics on X are homeomorphic to ℓ2 if X is compact, and to ℓ∞ if X is not compact.
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