{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:48:39Z","timestamp":1760150919450,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,27]],"date-time":"2022-01-27T00:00:00Z","timestamp":1643241600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The structure of topological spaces is analysed here through the lenses of fibrous preorders. Each topological space has an associated fibrous preorder and those fibrous preorders which return a topological space are called spatial. A special class of spatial fibrous preorders consisting of an interconnected family of preorders indexed by a unitary magma is called Cartesian and is studied here. Topological spaces that are obtained from those fibrous preorders, with a unitary magma I, are called I-Cartesian and are characterized. The characterization reveals a hidden structure on such spaces. Several other characterizations are obtained, and special attention is drawn to the case of a monoid equipped with a topology. A wide range of examples is provided, as well as general procedures to obtain topologies from other data types such as groups and their actions. Metric spaces and normed spaces are considered as well.<\/jats:p>","DOI":"10.3390\/axioms11020049","type":"journal-article","created":{"date-parts":[[2022,1,27]],"date-time":"2022-01-27T21:59:55Z","timestamp":1643320795000},"page":"49","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Structure of Topological Spaces"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4199-7367","authenticated-orcid":false,"given":"Nelson","family":"Martins-Ferreira","sequence":"first","affiliation":[{"name":"School of Technology and Management, Polit\u00e9cnico de Leiria, 2411-901 Leiria, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,27]]},"reference":[{"key":"ref_1","unstructured":"James, I.M. (1999). History of Topology, North-Holland."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.hm.2008.01.001","article-title":"The emergence of open sets, closed sets, and limit points in analysis and topology","volume":"35","author":"Moore","year":"2008","journal-title":"Hist. Math."},{"key":"ref_3","unstructured":"Fricke, R., Noether, E., and Ore, \u00d6. (1931). Gesammelte Mathematische Werke, Vieweg."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1007\/BF02420592","article-title":"Int\u00e9grale, longueur, aire","volume":"7","author":"Lebesgue","year":"1902","journal-title":"Ann. Mat. Pura Appl."},{"key":"ref_5","first-page":"224","article-title":"Sur un th\u00e9or\u00e8me de M. Borel","volume":"140","author":"Riesz","year":"1905","journal-title":"Comptes Rendus Hebd. Des Seances De L\u2019academie Des Sci. Paris"},{"key":"ref_6","unstructured":"de la Vall\u00e9e Poussin, C. (1916). Int\u00e9grales de Lebesgue, Fonctions d\u2019ensemble, Classes de Baire, Gauthier\u2013Villars."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF03018603","article-title":"Sur quelques points du Calcul fonctionnel","volume":"22","year":"1906","journal-title":"Rend. Circ. Mat. Palermo"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"341","DOI":"10.24033\/asens.735","article-title":"Sur les ensembles abstraits","volume":"38","year":"1921","journal-title":"Ann. Sci. l\u2019Ecole Norm. Sup\u00e9rieure"},{"key":"ref_9","unstructured":"Hausdorff, F. (1914). Grundz\u00fcge der Mengenlehre, Veit."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Hausdorff, F. (1927). Mengenlehre, de Gruyter.","DOI":"10.1515\/crll.1927.158.113"},{"key":"ref_11","unstructured":"Carath\u00e9odory, C. (1918). Vorlesungen \u00fcber Reelle Funktionen, Teubner."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"182","DOI":"10.4064\/fm-3-1-182-199","article-title":"Sur l\u2019op\u00e9ration A\u00af de l\u2019analysis situs","volume":"3","author":"Kuratowski","year":"1922","journal-title":"Fundam. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1007\/BF01579182","article-title":"Beitr\u00e4ge zur allgemeinen Topologie. I. Axiome f\u00fcr verschiedene Fassungen des Umgebungsbegriffs","volume":"88","author":"Tietze","year":"1923","journal-title":"Math. Ann."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1007\/BF01208660","article-title":"Zur Begr\u00fcndung der n-dimensionalen mengentheoretischen Topologie","volume":"94","author":"Aleksandrov","year":"1925","journal-title":"Math. Ann."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Aleksandrov, P., and Hopf, H. (1935). Topologie, Springer.","DOI":"10.1007\/978-3-662-02021-0"},{"key":"ref_16","unstructured":"Sierpinski, W. (1934). Introduction to General Topology, University of Toronto."},{"key":"ref_17","unstructured":"Kuratowski, K. (1933). Topologie I. Espaces M\u00e9trisables, Espaces Complets, Garasinski."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Lefschetz, S. (1930). Topology, American Mathematical Society.","DOI":"10.1090\/coll\/012"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Lefschetz, S. (1942). Algebraic Topology, American Mathematical Society.","DOI":"10.1090\/coll\/027"},{"key":"ref_20","unstructured":"Bourbaki, N. (1951). El\u00e9ments de Math\u00e9matique II. Premi\u00e8re Partie. Les Structures Fondamentales de l\u2019analyse Livre III. Topologie G\u00e9n\u00e9rale. Chapitre I. Structures Topologiques, Hermann. [2nd ed.]. Actualit\u00e9s Scientifiques et Industrielles, vol.1142."},{"key":"ref_21","unstructured":"Kelley, J.L. (1955). General Topology, Van Nostrand."},{"key":"ref_22","first-page":"501","article-title":"Diskrete R\u00e4ume","volume":"2","author":"Aleksandrov","year":"1937","journal-title":"Mat. Sbornik"},{"key":"ref_23","unstructured":"Montgomery, D., and Zippin, L. (1955). Topological Transformation Groups, Interscience Publishers."},{"key":"ref_24","unstructured":"Brown, R. (1988). Topology and Groupoids (or Topology, a Geometric Account of General Topology, Homotopy Types and the Fundamental Groupoid), Halsted Press."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Arhangel\u2019skii, A.V., and Tkachenko, M. (2008). Topological Groups and Related Structures, World Sci.","DOI":"10.2991\/978-94-91216-35-0"},{"key":"ref_26","unstructured":"Morris, S.A. (2019). Topological Groups Advances, Surveys, and Open Questions, MDPI. Printed Edition of the Special Issue Published in Axioms."},{"key":"ref_27","first-page":"296","article-title":"G-groupoids, crossed modules and the fundamental groupoid of a topological group","volume":"79","author":"Brown","year":"1976","journal-title":"Proc. Kon. Ned. Akad. Wet"},{"key":"ref_28","first-page":"221","article-title":"From A-spaces to arbitrary spaces via spatial fibrous preorders","volume":"46","year":"2014","journal-title":"Math Texts (Categ. Methods Algebra Topol.)"},{"key":"ref_29","first-page":"14","article-title":"The notion of multi-link, its applications and examples","volume":"7","year":"2016","journal-title":"Scripta-Ingenia"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/2\/49\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:08:40Z","timestamp":1760134120000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/2\/49"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,27]]},"references-count":29,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["axioms11020049"],"URL":"https:\/\/doi.org\/10.3390\/axioms11020049","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,1,27]]}}}