A result concerning the Lipschitz realcompactification of the product of two metric spaces
For a metric space (X, d), we consider the so-called Lipschitz realcompactification of X, denoted by H(Lipd(X)). In this note we give a result concerning the equality H(Lipd+ρ(X × Y )) = H(Lipd(X)) × H(Lipρ(Y )) for the product of the two metric spaces (X, d) and (Y, ρ). More precisely, we prove tha...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universidad Complutense de Madrid (UCM) |
| Repositorio: | Docta Complutense |
| Idioma: | inglés |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/128489 |
| Acceso en línea: | https://hdl.handle.net/20.500.14352/128489 |
| Access Level: | acceso abierto |
| Palabra clave: | Metric spaces Real-valued Lipschitz functions Lipschitz realcompactification Samuel compactification Topología 1210.05 Topología General |
| Sumario: | For a metric space (X, d), we consider the so-called Lipschitz realcompactification of X, denoted by H(Lipd(X)). In this note we give a result concerning the equality H(Lipd+ρ(X × Y )) = H(Lipd(X)) × H(Lipρ(Y )) for the product of the two metric spaces (X, d) and (Y, ρ). More precisely, we prove that such equality holds if and only if H(Lipd(X)) = X or H(Lipρ(Y )) = Y , where X and Y denote the completion of X and Y respectively, or equivalently, if and only if the Lipschitz realcompactification of one of the factors X or Y is as simple as possible. We also point out that our result is, in fact, a true generalization of a known theorem by Woods about the Samuel compactification of the product of two metric spaces. |
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