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[{"key": "dc.contributor.advisor", "value": "Rajala, Kai", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Varis, Valtteri", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2023-10-05T06:14:06Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2023-10-05T06:14:06Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2023", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/89389", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4 tutkielma on katsaus topologiaan keskittyen etenkin parakompaktiuteen ja avaruuksien metristyvyyteen. Tutkielmassa esitell\u00e4\u00e4n topologian perusteet avoimista joukoista alkaen ja t\u00e4m\u00e4n j\u00e4lkeen k\u00e4yd\u00e4\u00e4n l\u00e4pi tarvittavia esitietoja, kuten topologian kanta, jatkuva funktio ja separaatioaksioomat. Er\u00e4s t\u00e4rkeimmist\u00e4 esitiedoista on paikallisen \u00e4\u00e4rellisyyden k\u00e4site.\nParakompaktiutta vahvempi ominaisuus topologiselle avaruudelle on kompaktius. Kompaktin joukon voidaan ajatella olevan yleistys suljetusta ja rajoitetusta v\u00e4list\u00e4\nreaalilukujen joukossa. Tutkielmassa n\u00e4ytet\u00e4\u00e4n, miten kompaktiuden m\u00e4\u00e4ritelm\u00e4st\u00e4 saadaan muotoiltua parakompaktiuden m\u00e4\u00e4ritelm\u00e4 ja mit\u00e4 tuloksia parakompaktiuden nojalla topologiselle avaruudelle voidaan todistaa. Yksi n\u00e4ist\u00e4 tuloksista on metristyvyys, joka tarkoittaa, ett\u00e4 kyseisess\u00e4 topologisessa avaruudessa voidaan m\u00e4\u00e4ritell\u00e4 et\u00e4isyyden k\u00e4site. Metristyvyydelle on topologian historian aikana annettu useita ehtoja ja t\u00e4ss\u00e4 tutkielmassa niist\u00e4 esitet\u00e4\u00e4n kolme. Kaksi ensimm\u00e4ist\u00e4 n\u00e4ist\u00e4, eli Urysonin metristyvyyslause ja Nagata-Smirnovin metristyvyyslause, eiv\u00e4t k\u00e4yt\u00e4 todistuksissaan parakompaktiutta. Viimeisen\u00e4 esitett\u00e4v\u00e4 Smirnovin metristyvyyslause taas k\u00e4ytt\u00e4\u00e4 parakompaktiutta.\nTutkielmassa k\u00e4sitell\u00e4n useita parakompaktiuden sovelluksia. Smirnovin metristyvyyslauseen lis\u00e4ksi tarkastellaan monistoja, joille saadaan parakompaktiutta k\u00e4ytt\u00e4en osoitettua erilaisia ominaisuuksia. Monistot ovat topologisia avaruuksia, jotka paikallisesti n\u00e4ytt\u00e4v\u00e4t euklidisilta avaruuksilta. Esimerkiksi fysiikassa aika-avaruutta voidaan mallintaa monistona. Parakompaktiuden sovellukset ovat usein seurausta ykk\u00f6sen osituksen olemassaolosta. Ykk\u00f6sen osituksessa laajennetaan paikallisesti m\u00e4\u00e4riteltyj\u00e4 jatkuvia funktioita koko avaruuteen.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2023-10-05T06:14:06Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2023-10-05T06:14:06Z (GMT). No. of bitstreams: 0\n Previous issue date: 2023", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "32", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": null, "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "metristyvyys", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Parakompaktius", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202310055419", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.copyright", "value": "\u00a9 The Author(s)", "language": null, "element": "rights", "qualifier": "copyright", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "topologia", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "topologiset avaruudet", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "monistot", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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