S-Metrizability and the Wallman basis of a frame | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 7، دوره 20، شماره 1، فروردین 2024، صفحه 155-174 اصل مقاله (493 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.48308/cgasa.2023.233801.1440 | ||
| نویسنده | ||
| Cerene Rathilal* | ||
| Univeristy of KwaZulu-Natal | ||
| چکیده | ||
| The Wallman basis of a frame and the corresponding induced compactification was first investigated by Baboolal [2]. In this paper, we provide an intrinsic characterisation of S-metrizability in terms of the Wallman basis of a frame. Particularly, we show that a connected, locally connected frame is S-metrizable if and only if it has a countable locally connected and uniformly connected Wallman basis. | ||
| کلیدواژهها | ||
| S-metrizable؛ Wallman basis؛ compactification؛ frame | ||
| مراجع | ||
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[1] Baboolal, B., Connectedness in metric frames, Appl. Categ. Structures 13 (2005), 161-169. [2] Baboolal, B., Local connectedness and the Wallman compactification, Quaestions Mathematicae. (2012), 245-257. [3] Baboolal, D. and Banaschewski, B., Compactification and local connectedness of frames, Pure Appl. Algebra 70 (1991), 3-16. [4] Banaschewski, B., Compactification of frames, Math. Nachr. 149 (1990), 105-116. [5] Banaschewski, B., Lecture on Frames, Seminar, University of Cape Town, 1988. [6] Banaschewski, B. and Pultr, A., A Stone Duality for Metric Spaces, American Mathematical Society, 1992. [7] Garc´ıa-M´aynez, A., Property C, Wallman basis and S-metrizability, Topology Appl. 12 (1981), 237-246. [8] Johnstone, P.T., Wallman compactification of locales, Houston J. Math. 10(2) (1984), 201-206.[9] Picado, J. and Pultr, A., Frames and Locales: Topology without points, Frontiers in Mathematics, Springer Basel AG, 2012. [10] Pultr, A., Pointless uniformities I. complete regularity, Comment. Math. Univ. Carolin 25(1) (1984), 91-104. [11] Rathilal, C., A note on S-metrizable frames, Topology Appl 307 (2022). [12] Sierpinski, W., Sur une condition pour qu’un continu soit une courbe jordanienne, Fund. Math 1 (1920), 44-60. [13] Steiner, E.F., Wallman spaces and compactification, Fund. Math. 61 (1967), 295-304. | ||
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