{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T11:35:53Z","timestamp":1740137753804,"version":"3.37.3"},"reference-count":18,"publisher":"EDP Sciences","license":[{"start":{"date-parts":[[2024,2,19]],"date-time":"2024-02-19T00:00:00Z","timestamp":1708300800000},"content-version":"vor","delay-in-days":49,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Theor. Inf. Appl."],"accepted":{"date-parts":[[2024,1,16]]},"published-print":{"date-parts":[[2024]]},"abstract":"<jats:p>Infinite words are often considered as limits of finite words. As topological methods have been proved to be useful in the theory of <jats:italic>\u03c9<\/jats:italic>-languages it seems to be providing to include finite and infinite words into one (topological) space. In most cases this results in a poor topological structure induced on the subspace of finite words. In the present paper we investigate the possibility to link topologies in the space of finite words with a topology in the space of infinite words <jats:italic>via<\/jats:italic> a natural mapping. A requirement in this linking of topologies consists in the compatibility of the topological properties (openness, closedness <jats:italic>etc<\/jats:italic>.) of images with preimages and vice versa. Here, we show that choosing for infinite words the natural topology of the CANTOR space and the <jats:italic>\u03b4<\/jats:italic>-limit as linking mapping there are several natural topologies on the space of finite words compatible with the topology of the CANTOR space. It is interesting to observe that besides the well-known prefix topology there are at least two more whose origin is from language theory \u2013 centers and supercenters of languages. We show that several of these topologies on the space of finite words fit into a class of \u2112-topologies and exhibit their special properties w.r.t. to the compatibility with the CANTOR topology.<\/jats:p>","DOI":"10.1051\/ita\/2024002","type":"journal-article","created":{"date-parts":[[2024,2,19]],"date-time":"2024-02-19T08:52:55Z","timestamp":1708332775000},"page":"2","source":"Crossref","is-referenced-by-count":0,"title":["Topologies for Finite Words: Compatibility with the Cantor Topology"],"prefix":"10.1051","volume":"58","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3810-9303","authenticated-orcid":false,"given":"Ludwig","family":"Staiger","sequence":"first","affiliation":[]}],"member":"250","published-online":{"date-parts":[[2024,2,19]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0022-0000(80)90010-0","volume":"20","author":"Boasson","year":"1980","journal-title":"Adherences of languages. J. Comput. Syst. 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Springer, Berlin (1997) 339\u2013387.","DOI":"10.1007\/978-3-642-59126-6_6"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"Staiger L., Joint topologies for finite and infinite words, in Developments in Language Theory, Vol. 6224 of Lecture Notes in Computer Science, edited by Gao Y., Lu H., Seki S. and Yu S.. Springer (2010) 442\u2013443.","DOI":"10.1007\/978-3-642-14455-4_44"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"Staiger L. and Nehrlich W., The centers of context-sensitive languages, in Mathematical Foundations of Computer Science, Vol. 233 of Lecture Notes in Comput. Sci., edited by Gruska J., Rovan B. and Wiedermann J.. Springer, Berlin (1986) 594\u2013601.","DOI":"10.1007\/BFb0016286"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"Thomas W., Automata on infinite objects, in Handbook of Theoretical Computer Science, Vol. B, edited by van Leeuwen J.. 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