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[{"key": "dc.contributor.advisor", "value": "Koskela, Pekka", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Timonen, Joel", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2023-06-27T05:41:00Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2023-06-27T05:41:00Z", "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/123456789/88075", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 matematiikan Pro Gradu -tutkielmassa todistetaan McShanen ja Tietzen jatkolauseet sek\u00e4 Urysonin lemma. Ensimm\u00e4inen tulos liittyy metrisiin avaruuksiin ja kaksi j\u00e4lkimm\u00e4ist\u00e4 topologiaan.\nMcShanen jatkolause kertoo, ett\u00e4 metrisen avaruuden osajoukossa m\u00e4\u00e4ritelty tasaisesti jatkuva kuvaus voidaan laajentaa jatkuvaksi koko avaruuteen. T\u00e4m\u00e4n lauseen yhteydess\u00e4 oleellinen k\u00e4site on kuvauksen jatkuvuusmoduli, joka antaa kvantitatiivisen tavan k\u00e4sitell\u00e4 tasaista jatkuvuutta. Tietyin lis\u00e4edellytyksin McShanen jatkolause kertoo, ett\u00e4 kuvaus voidaan laajentaa koko avaruuteen siten, ett\u00e4 laajennuksella on sama jatkuvuusmoduli kuin alkuper\u00e4isell\u00e4 kuvauksella. McShanen jatkolauseen todistuksessa tarvitaan joitain konveksianalyysin tuloksia. Niinp\u00e4 t\u00e4h\u00e4nkin matematiikan osa-alueeseen tutustutaan, tosin hyvin pintapuolisesti.\nTietzen jatkolause puolestaan kertoo, ett\u00e4 normaalin topologisen avaruuden suljetussa osajoukossa m\u00e4\u00e4ritelty jatkuva kuvaus voidaan laajentaa jatkuvaksi koko avaruuteen. T\u00e4m\u00e4n tuloksen todistamisessa avaruuden normaalius on oleellista, joten t\u00e4ss\u00e4 tutkielmassa tutustutaan my\u00f6s separaatioaksioomiin (joihin normaalius liittyy l\u00e4heisesti). Urysonin lemma kertoo, ett\u00e4 normaalissa avaruudessa kahden erillisen suljetun joukon v\u00e4lill\u00e4 on olemassa jatkuva kuvaus, joka saa yhdess\u00e4 joukossa arvon 0 ja toisessa arvon 1; toisin sanoen normaalin avaruuden erilliset suljetut osajoukot voidaan erotella toisistaan jatkuvan kuvauksen avulla. Urysonin lemmaa k\u00e4ytet\u00e4\u00e4n apuna Tietzen jatkolauseen todistamisessa.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2023-06-27T05:41:00Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2023-06-27T05:41:00Z (GMT). No. of bitstreams: 0\n Previous issue date: 2023", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "39", "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": "separaatioaksioomat", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Separaatioaksioomat ja jatkuvien kuvausten laajentaminen", "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-202306274136", "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": "metriset avaruudet", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "jatkuvuus", "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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