Rathilal, Cerene A note on S-metrizable frames. (English) Zbl 1508.06007 Topology Appl. 307, Article ID 107951, 10 p. (2022). Summary: A metric frame \((M,d)\) is S-metrizable if there exists a compatible metric diameter \(\rho\) on \(M\) such that \((M,\rho)\) has property S. The latter concept of property S was introduced to frames by D. Baboolal [Appl. Categ. Struct. 8, No. 1–2, 377–390 (2000; Zbl 0969.54004)]. Given a connected, locally connected metric frame \((M,d)\), we show that \((M,d)\) is S-metrizable if and only if \((M,d)\) has a perfect locally connected metrizable compactification. MSC: 06D22 Frames, locales 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54E35 Metric spaces, metrizability Keywords:dense metric sublocale; metric frame; metric frame completion; perfect compactification; property S; S-metrizable Citations:Zbl 0969.54004 PDFBibTeX XMLCite \textit{C. Rathilal}, Topology Appl. 307, Article ID 107951, 10 p. (2022; Zbl 1508.06007) Full Text: DOI References: [1] Baboolal, D.; Banaschewski, B., Compactification and local connectedness of frames, J. Pure Appl. Algebra, 70, 3-16 (1991) · Zbl 0722.54031 [2] Baboolal, D., Local connectedness made uniform, Appl. Categ. Struct., 8, 377-390 (2000) · Zbl 0969.54004 [3] Baboolal, D., Connectedness in metric frames, Appl. Categ. Struct., 13, 161-169 (2005) · Zbl 1078.54002 [4] Baboolal, D., Local connectedness and the Wallman compactification, Quaest. Math., 35, 245-257 (2012) · Zbl 1274.06037 [5] Banaschewski, B.; Pultr, A., Samuel compactification and the completion of uniform frames, Math. Proc. Camb. Philos. Soc., 108, 1, 63-78 (1990) · Zbl 0733.54020 [6] Banaschewski, B.; Pultr, A., A Stone Duality for Metric Spaces (1992), American Mathematical Society · Zbl 0789.54035 [7] García-Máynez, A., Property C, Wallman basis and S-metrizability, Topol. Appl., 12, 3, 237-246 (1981) · Zbl 0456.54012 [8] Picado, J.; Pultr, A., Frames and Locales: Topology Without Points, Frontiers in Mathematics, vol. 28 (2012), Springer: Springer Basel · Zbl 1231.06018 [9] Pultr, A., Diameters in locales: how bad can they be, Comment. Math. Univ. Carol., 29 (1988) · Zbl 0668.06008 [10] Pultr, A., Pointless uniformities II. (Dia)metrization, Comment. Math. Univ. Carol., 25, 105-120 (1984) · Zbl 0543.54023 [11] Rathilal, C.; Baboolal, D.; Pillay, P., Property S and local connectedness in metric frames, Topol. Appl., 267, Article 106888 pp. (2019) · Zbl 1468.54022 [12] Sierpinski, W., Sur une condition pour qu’un continu soit une courbe Jordanienne, Fundam. Math., 1, 44-60 (1920) · JFM 47.0522.02 [13] Whyburn, G. T., A certain transformation on metric spaces, Am. J. Math., 54, 2, 367-376 (1932) · JFM 58.0635.04 [14] Whyburn, G. T., Analytic Topology, Amer. Math. Soc. Colloq. Publ., vol. 28 (1942) · Zbl 0061.39301 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.