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2 Bulletin of the Polish Academy of Sciences. Mathematics

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Recognizing the topologies of spaces of metrics with the topology of uniform convergence

Koshino Katsuhisa

Bulletin of the Polish Academy of Sciences. Mathematics

2022

Vol. 70, no. 2

165--171

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.

On linear operators extending [pseudo]metrics

Banach T., Bessaga C.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 1

35-49

For every closed subset X of a stratifiable [respectively metrizable] space Y we construct a positive linear extension operator T : R[sup X*X] --> R[sup Y*Y] preserving constant functions, bounded functions, continuous functions, pseudometrics, metrics, [respectively dominating metrics, and admissible metrics]. This operator is continuous with respect to each of the three topologies : point-wise convergence, uniform, and compact-open. An equivariant analog of the above statement is proved as well.