{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:58:04Z","timestamp":1760234284302,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,5,1]],"date-time":"2021-05-01T00:00:00Z","timestamp":1619827200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The interplay of symmetry of algebraic structures in a space and the corresponding topological properties of the space provides interesting insights. This paper proposes the formation of a predicate evaluated P-separation of the subspace of a topological (C, R) space, where the P-separations form countable and finite number of connected components. The Noetherian P-separated subspaces within the respective components admit triangulated planar convexes. The vertices of triangulated planar convexes in the topological (C, R) space are not in the interior of the Noetherian P-separated open subspaces. However, the P-separation points are interior to the respective locally dense planar triangulated convexes. The Noetherian P-separated subspaces are surjectively identified in another topological (C, R) space maintaining the corresponding local homeomorphism. The surjective identification of two triangulated planar convexes generates a quasiloop\u2013quasigroupoid hybrid algebraic variety. However, the prime order of the two surjectively identified triangulated convexes allows the formation of a cyclic group structure in a countable discrete set under bijection. The surjectively identified topological subspace containing the quasiloop\u2013quasigroupoid hybrid admits linear translation operation, where the right-identity element of the quasiloop\u2013quasigroupoid hybrid structure preserves the symmetry of distribution of other elements. Interestingly, the vertices of a triangulated planar convex form the oriented multiplicative group structures. The surjectively identified planar triangulated convexes in a locally homeomorphic subspace maintain path-connection, where the right-identity element of the quasiloop\u2013quasigroupoid hybrid behaves as a point of separation. Surjectively identified topological subspaces admitting multiple triangulated planar convexes preserve an alternative form of topological chained intersection property.<\/jats:p>","DOI":"10.3390\/sym13050783","type":"journal-article","created":{"date-parts":[[2021,5,1]],"date-time":"2021-05-01T21:35:39Z","timestamp":1619904939000},"page":"783","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Surjective Identifications of Convex Noetherian Separations in Topological (C, R) Space"],"prefix":"10.3390","volume":"13","author":[{"given":"Susmit","family":"Bagchi","sequence":"first","affiliation":[{"name":"Department of Aerospace and Software Engineering (Informatics), Gyeongsang National University, Jinju 660701, Korea"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3957","DOI":"10.1073\/pnas.84.12.3957","article-title":"A topological concept of smallness","volume":"84","author":"Pfeffer","year":"1987","journal-title":"Proc. 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