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On linear operators extending [pseudo] metrics. (English) Zbl 0948.54021

The authors prove the following two interesting results and pose two questions.
Theorem 1. Suppose \(Y\) is a stratifiable space and \(X\) is a closed space of \(Y\) with \(|X|\geq 2\). There exists a positive linear extension operator \(T: \mathbb{R}^{X\times X}\to \mathbb{R}^{Y\times Y}\) preserving constant functions, bounded functions, continuous functions, pseudometrics, and metrics. This operator is continuous with respect to each of the three topologies: pointwise convergence, uniform, and compact-open.
Theorem 2. Suppose \(Y\) is a metrizable uniform space and \(X\) is a closed subspace of \(Y\) with \(|X|\geq 2\). There exists a positive linear extension operator \(T: \mathbb{R}^{X\times X}\to \mathbb{R}^{Y\times Y}\) preserving constant functions, bounded functions, continuous functions, pseudometrics, metrics, admissible metrics, dominating metrics, and uniformly dominating metrics. This operator is continuous with respect to each of the three topologies: pointwise convergence, uniform, and compact open. Moreover, if the uniform space \(Y\) is complete, then \(T\) preserves complete continuous uniformly dominating metrics. If \(Y\) is totally bounded and \(\dim (Y\smallsetminus X)<\infty\), then \(T\) preserves totally bounded pseudometrics.

MSC:

54C20 Extension of maps
54E20 Stratifiable spaces, cosmic spaces, etc.
54C35 Function spaces in general topology
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