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[{"key": "dc.contributor.advisor", "value": "Ilmavirta, Joonas", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Salo, Mikko", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2017-06-06T16:51:56Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2017-06-06T16:51:56Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2017", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1703144", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/54335", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa k\u00e4sitell\u00e4\u00e4n topologisia avaruuksia ja erityisesti niiden kompaktisuutta. Topologiset avaruudet ovat yleistys normiavaruuksista, mutta niiss\u00e4 ei tunneta et\u00e4isyyden k\u00e4sitett\u00e4. Topologisia k\u00e4sitteit\u00e4 ovatkin sellaiset, jotka s\u00e4ilyv\u00e4t avaruuden jatkuvissa muodonmuutoksissa, kuten venytyksiss\u00e4 ja taivutuksissa. Topologian n\u00e4k\u00f6kulmasta esimerkiksi v\u00e4li $(0,1)$ on sama kuin koko reaaliakseli $\\mathbb{R}$.\n\nKompaktisuus on yksi t\u00e4rkeimpi\u00e4 topologisia ominaisuuksia ja tutkielmassa todistetaankin useita kompaktisuuteen liittyvi\u00e4 tuloksia, joista t\u00e4rkein on ehdottomasti Tihonovin lause. Tihonovin lauseen sovelluksena todistamme my\u00f6s Heine-Borelin lauseen, joka karakterisoi euklidisen avaruuden $\\mathbb{R}^n$ kompaktit osajoukot. Kompaktilla avaruudella on monia hy\u00f6dyllisi\u00e4 ja haluttuja ominaisuuksia. Tunnettuna esimerkkin\u00e4 n\u00e4ist\u00e4 on se, ett\u00e4 jatkuva kuvaus $f\\colon \\mathbb{R}\\to \\mathbb{R}$ saavuttaa suurimman ja pienemmin arvonsa jokaisessa kompaktissa joukossa.\n\nLis\u00e4ksi tutkielmassa perehdyt\u00e4\u00e4n siihen, miten topologista avaruutta approksimoidaan kompaktilla topologisella avaruudella. T\u00e4t\u00e4 kutsutaan kompaktisoinniksi ja se tapahtuu upottamalla topologinen avaruus kompaktiin topologiseen avaruuteen siten, ett\u00e4 alkuper\u00e4inen avaruus on topologian mieless\u00e4 hyvin suuri uudessa kompaktissa avaruudessa. Esimerkkin\u00e4 kompaktisoinnista annetaan yhden pisteen kompaktisointi ja Stone-\u010cech-kompaktisointi, jotka tullaan osoittamaan tietyiss\u00e4 tapauksissa pienimm\u00e4ksi ja suurimmaksi kompaktisoinniksi.\n\nEsitiedoiksi lukijalta vaaditaan perustaidot joukko-opista. Lis\u00e4ksi Heine-Borelin lauseen ymm\u00e4rt\u00e4miseen vaaditaan tietoja vektoriavaruuksista ja erityisesti euklidisista avaruuksista.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "In this thesis we address topological spaces and especially their compactness. Topological spaces are a generalization of inner product spaces, but they don't have the concept of distance. Topological concepts are those that are preserved under continuous deformations, such as stretching and bending. For example in topological terms the interval $(0,1)$ is the same as the entire real axis $\\mathbb{R}$\n\nOne of the most important topological concepts is compactness and we prove many theorems regarding compactness, the most important of which is by far Tychonoff theorem. As an application of Tychonoff theorem we also prove Heine-Borel theorem which characterizes the compact subsets of the euclidean space $\\mathbb{R}^n$. Compact spaces have many useful and wanted properties. A well known example of these is that a continuous mapping $f\\colon \\mathbb{R}\\to \\mathbb{R}$ attains its minimal and maximal values on any compact interval.\n\nAdditionally in this thesis we look into approximating topological spaces with compact topological spaces. This is called compactification and it's done by embedding a topological space into a compact topological space such that the original space is very large in the new compact space in terms of topology. As examples of compactification we give the one point compactification and Stone-\u010cech-compactification, which turn out to be the smallest and largest compactifications in certain cases.\n\nThe reader is expected to have basic understanding of set theory. Additionally, in order to understand the Heine-Borel theorem, the reader must have knowledge of vector spaces, especially of the euclidean space.", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Mikko Salo (mivasalo) on 2017-06-06 16:51:56.433666. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2017-06-06T16:51:56Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201706062710.pdf: 269388 bytes, checksum: 6cc64191c2d940aa395b9335637ce2b6 (MD5)\nlicense.html: 4779 bytes, checksum: dd60324af3a2da16d44312bd396a00c5 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2017-06-06T16:51:56Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201706062710.pdf: 269388 bytes, checksum: 6cc64191c2d940aa395b9335637ce2b6 (MD5)\nlicense.html: 4779 bytes, checksum: dd60324af3a2da16d44312bd396a00c5 (MD5)\n Previous issue date: 2017", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (53 sivua)", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "topologia", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "kompaktisointi", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Kompaktisuus ja kompaktisointi", "language": null, "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-201706062710", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "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": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.date.updated", "value": "2017-06-06T16:51:57Z", "language": null, "element": "date", "qualifier": "updated", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": null, "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.accesslevel", "value": "openAccess", "language": "fi", "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": null, "element": "subject", "qualifier": "oppiainekoodi", "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.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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