{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:44:16Z","timestamp":1759970656024,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,1,29]],"date-time":"2025-01-29T00:00:00Z","timestamp":1738108800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU)","award":["IMSIU-RG23137"],"award-info":[{"award-number":["IMSIU-RG23137"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let \u03c4 be a topology on the finite set Xn. We consider the open-set polynomial associated with the topology \u03c4. Its coefficients are the cardinalities of sets Uj=Uj(\u03c4) of open sets of size j=0,\u2026,n. We prove that this polynomial has only real zeros only in the trivial case where \u03c4 is the discrete topology. Hence, we answer a question raised by J. Brown. We give a partial answer to the question: for which topology is this polynomial log-concave, or at least unimodal? More specifically, we prove that if the topology has a large number of open sets, its open polynomial is unimodal. The idea of degree of log-concavity is introduced and it is shown to be limited for polynomials of non-trivial topologies. Furthermore, the maximum-sized topologies that omit open sets of given sizes are derived. Moreover, all topologies over n points with at least (3\/8)2n open sets are proved to be unimodal, completing previous results.<\/jats:p>","DOI":"10.3390\/axioms14020103","type":"journal-article","created":{"date-parts":[[2025,1,29]],"date-time":"2025-01-29T07:45:12Z","timestamp":1738136712000},"page":"103","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Polynomials Associated with Finite Topologies"],"prefix":"10.3390","volume":"14","author":[{"given":"Moussa","family":"Benoumhani","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11586, Saudi Arabia"}]},{"given":"Brahim","family":"Chaourar","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11586, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,29]]},"reference":[{"key":"ref_1","first-page":"73","article-title":"History of the number of finite posets","volume":"5","author":"Klaska","year":"1997","journal-title":"Acta Univ. 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Handbook of Enumerative Combinatorics, CRC Press."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/0166-218X(88)90017-0","article-title":"On some counting polynomials","volume":"19","author":"Hosoya","year":"1988","journal-title":"Discret. Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/s00026-005-0255-5","article-title":"Bounding the Roots of Ideal and open-set polynomials","volume":"9","author":"Brown","year":"2005","journal-title":"Ann. Comb."},{"key":"ref_13","unstructured":"Brown, J. (2008). On the Zeros of Independence and Open Polynomials, Isaac Newton Institute. Seminar."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1111\/j.1749-6632.1989.tb16434.x","article-title":"Log-concave and unimodal sequences in algebra, combinatorics, and geometry","volume":"576","author":"Stanley","year":"1989","journal-title":"Ann. N. Y. Acad. Sci."},{"key":"ref_15","unstructured":"Merrifield, R.E., and Simmons, H.E. (1989). Topological Methods in Chemistry, Wiley."},{"key":"ref_16","first-page":"07.3.1","article-title":"Direct and elementary approach to enumerate topologies on finite set","volume":"10","author":"Kolli","year":"2007","journal-title":"J. Integer Seq."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/0097-3165(71)90065-3","article-title":"On the Number of Open Sets of Finite Topologies","volume":"10","author":"Stanley","year":"1971","journal-title":"J. Comb. Theory"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/103\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:38:17Z","timestamp":1759919897000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/103"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,29]]},"references-count":17,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["axioms14020103"],"URL":"https:\/\/doi.org\/10.3390\/axioms14020103","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,1,29]]}}}