tag:blogger.com,1999:blog-922111353248289998Tue, 08 Sep 2026 13:00:20 +0000Definitions and PropositionsInformation TablesExploiting an Open-Source Office SuiteThe Bias PlanetTo Develop UNO Extensions (LibreOffice Extensions or Apache OpenOffice Extensions)Let Me Understand C++Let Me Understand the Python Programming LanguageJava TipsSchool Mathematics from Higher ViewpointsTo Disentangle Confusing Terms or DiscoursesLet Me Understand GradleLet Me Understand the Java Programming LanguageLet Me Understand C#Let Me Understand GitProjects Build SystemsGradle TipsHow to Use UNO (Handle LibreOffice or Apache OpenOffice Documents) in External Java ProgramsNotes About Using UNO in Basic MacrosUNO Dispatch Commandshttps://thebiasplanet.blogspot.com/noreply@blogger.com (Unknown)Blogger2463125tag:blogger.com,1999:blog-922111353248289998.post-4139639639012672262Sun, 06 Sep 2026 13:31:35 +00002026-09-06T22:31:35.585+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | description/proof of that for group, normal subgroup, and element of group, left coset of subgroup by element is right coset of subgroup by element Topics About: group The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Proofhttps://thebiasplanet.blogspot.com/2026/09/forgroupnormalsubgroupandelementofgroupleftcosetofsubgroupbyelementisrightcosetofsubgroupbyelement.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1207384185527869112Sun, 06 Sep 2026 13:30:10 +00002026-09-06T22:30:10.186+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for group with topology with continuous operations (especially, topological group) and closed subgroup, cosets of subgroup quotient topological space is Hausdorff and classification map is open Topics About: topological space The https://thebiasplanet.blogspot.com/2026/09/forgroupwithtopologywithcontinuousoperationsespeciallytopologicalgroupandclosedsubgroupcosetsofsubgroupquotienttopologicalspaceishausdorffandclassificationmapisopen.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1025804235975125930Sun, 06 Sep 2026 13:28:45 +00002026-09-06T22:28:45.362+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> definition of left or right cosets of subgroup quotient topological space of group with topology with continuous operations (especially, topological group) Topics About: topological space The table of contents of this article Starting Context Target Context https://thebiasplanet.blogspot.com/2026/09/leftorrightcosetsofsubgroupquotienttopologicalspaceofgroupwithtopologywithcontinuousoperationsespeciallytopologicalgroup.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3843929543034292571Sun, 06 Sep 2026 13:27:14 +00002026-09-06T22:27:14.287+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for group, \(2\) elements, and subgroup, cosets of subgroup by elements are same iff product of inverse of element and element is contained in subgroup Topics About: group The table of contents of this article Starting Context Targethttps://thebiasplanet.blogspot.com/2026/09/forgroup2elementsandsubgroupcosetsofsubgroupbyelementsaresameiffproductofinverseofelementandelementiscontainedinsubgroup.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-846328714769277047Sun, 06 Sep 2026 13:25:45 +00002026-09-06T22:25:45.718+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for group, subgroup, and \(2\) subsets, if intersection of 1st subset and product of 2nd subset and subgroup is empty, intersection of product of 1st subset and subgroup and product of 2nd subset and subgroup is empty Topics About: group https://thebiasplanet.blogspot.com/2026/09/forgroupsubgroupand2subsetsifintersectionof1stsubsetandproductof2ndsubsetandsubgroupisemptyintersectionofproductof1stsubsetandsubgroupandproductof2ndsubsetandsubgroupisempty.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3719681368474338384Sun, 06 Sep 2026 13:24:23 +00002026-09-06T22:24:23.992+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for group, \(2\) subsets, and symmetric subset, if intersection of 1st subset and product of 2nd subset and symmetric subset is empty, intersection of product of 1st subset and symmetric subset and 2nd subset is empty Topics About: group https://thebiasplanet.blogspot.com/2026/09/forgroup2subsetsandsymmetricsubsetifintersectionof1stsubsetandproductof2ndsubsetandsymmetricsubsetisemptyintersectionofproductof1stsubsetandsymmetricsubsetand2ndsubsetisempty.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2737976067815639461Sun, 06 Sep 2026 13:22:42 +00002026-09-06T22:22:42.493+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for group, product of element and intersection of subsets is intersection of products of element and subsets Topics About: group The table of contents of this article Starting Context Target Context Orientation Main Body 1: https://thebiasplanet.blogspot.com/2026/09/forgroupproductofelementandintersectionofsubsetsisintersectionofproductsofelementandsubsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8778778256708748564Sun, 06 Sep 2026 13:20:37 +00002026-09-06T22:20:37.085+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for probability space, independent indexed set of sub-\(\sigma\)-algebras, and independent indexed set of measurable maps w.r.t. each sub-\(\sigma\)-algebra, indexed set of all maps is independent Topics About: measure space The tablehttps://thebiasplanet.blogspot.com/2026/09/forprobabilityspaceindependentindexedsetofsubsigmaalgebrasandindependentindexedsetofmeasurablemapswrteachsubsigmaalgebraindexedsetofallmapsisindependent.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6823387827158963301Sun, 06 Sep 2026 13:18:53 +00002026-09-06T22:18:53.492+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for finite indexed set of finite sets, disjointed union of indexed set, and commutative ring, product by disjointed union is product by index after products by elements of indexed set Topics About: ring The table of contents of this https://thebiasplanet.blogspot.com/2026/09/forfiniteindexedsetoffinitesetsdisjointedunionofindexedsetandcommutativeringproductbydisjointedunionisproductbyindexafterproductsbyelementsofindexedset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1210431461329515569Sun, 06 Sep 2026 13:17:25 +00002026-09-06T22:17:25.321+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for indexed set of sets and disjointed union of indexed set, intersection by disjointed union is intersection by index after intersections by elements of indexed set Topics About: set The table of contents of this article Starting https://thebiasplanet.blogspot.com/2026/09/forindexedsetofsetsanddisjointedunionofindexedsetintersectionbydisjointedunionisintersectionbyindexafterintersectionsbyelementsofindexedset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-219145001213256404Sun, 06 Sep 2026 13:16:02 +00002026-09-06T22:16:02.752+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> definition of disjointed union of indexed set of sets Topics About: set The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note Starting Context The reader knows a definition of https://thebiasplanet.blogspot.com/2026/09/disjointedunionofindexedsetofsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1317994886753810975Sun, 06 Sep 2026 13:14:44 +00002026-09-06T22:14:44.365+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that indexed set of measurable maps from probability space into same measurable space is independent iff indexed set of sub-\(\sigma\)-algebras induced by maps is independent Topics About: measure space The table of contents of this https://thebiasplanet.blogspot.com/2026/09/indexedsetofmeasurablemapsfromprobabilityspaceintosamemeasurablespaceisindependentiffindexedsetofsubsigmaalgebrasinducedbymapsisindependent.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8483934212832239422Sun, 06 Sep 2026 13:13:17 +00002026-09-06T22:13:17.507+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> definition of independent indexed set of measurable maps from probability space into same measurable space Topics About: measure space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: https://thebiasplanet.blogspot.com/2026/09/independentindexedsetofmeasurablemapsfromprobabilityspaceintosamemeasurablespace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3971049098644430484Sun, 06 Sep 2026 13:11:54 +00002026-09-06T22:11:54.346+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> definition of independent indexed set of sub-\(\sigma\)-algebras of probability space Topics About: measure space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note Starting Context https://thebiasplanet.blogspot.com/2026/09/independentindexedsetofsubsigmaalgebrasofprobabilityspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-66717351455439649Sun, 30 Aug 2026 14:29:09 +00002026-09-06T22:10:27.539+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that intersection of non-increasing sequence of nonempty open or closed subsets does not necessarily contain point Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Bodyhttps://thebiasplanet.blogspot.com/2026/08/intersectionofnonincreasingsequenceofnonemptyopenorclosedsubsetsdoesnotnecessarilycontainpoint.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3557206466446108077Sun, 30 Aug 2026 14:27:45 +00002026-08-30T23:27:45.349+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that union of complements of subsets is whole set iff intersection of subsets is empty Topics About: set The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Proof https://thebiasplanet.blogspot.com/2026/08/unionofcomplementsofsubsetsiswholesetiffintersectionofsubsetsisempty.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7713075902813790846Sun, 30 Aug 2026 14:26:18 +00002026-08-30T23:26:18.032+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that composition of homotopy equivalences is homotopy equivalence Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note 3: Proof https://thebiasplanet.blogspot.com/2026/08/compositionofhomotopyequivalencesishomotopyequivalence.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7407770843306734993Sun, 30 Aug 2026 14:24:52 +00002026-08-30T23:24:52.807+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for continuous map from 1st space into 2nd space and continuous map from 2nd space into 3rd space, if 1st map and composition of 2nd map after 1st map are homotopy equivalences, 2nd map is homotopy equivalence Topics About: topological space https://thebiasplanet.blogspot.com/2026/08/forcontinuousmapfrom1stspaceinto2ndspaceandcontinuousmapfrom2ndspaceinto3rdspaceif1stmapandcompositionof2ndmapafter1stmaparehomotopyequivalences2ndmapishomotopyequivalence.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3305031765601237268Sun, 30 Aug 2026 14:23:28 +00002026-08-30T23:23:28.832+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for continuous map from 1st space into 2nd space and continuous map from 2nd space into 3rd space, if 2nd map and composition of 2nd map after 1st map are homotopy equivalences, 1st map is homotopy equivalence Topics About: topological space https://thebiasplanet.blogspot.com/2026/08/forcontinuousmapfrom1stspaceinto2ndspaceandcontinuousmapfrom2ndspaceinto3rdspaceif2ndmapandcompositionof2ndmapafter1stmaparehomotopyequivalences1stmapishomotopyequivalence.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8981269666318872600Sun, 30 Aug 2026 14:22:11 +00002026-08-30T23:22:11.395+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for homotopy equivalence, map homotopic to homotopy equivalence is homotopy equivalence Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured https://thebiasplanet.blogspot.com/2026/08/forhomotopyequivalencemaphomotopictohomotopyequivalenceishomotopyequivalence.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5420132667702004501Sun, 30 Aug 2026 14:20:52 +00002026-08-30T23:20:52.343+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for \(2\) continuous maps with same domain and codomain and equivalence relations on domain and codomain, if each class is mapped into class and maps are homotopic relative to subset that contains multi-points classes, induced maps between quotient https://thebiasplanet.blogspot.com/2026/08/for2continuousmapswithsamedomainandcodomainandequivalencerelationsondomainandcodomainifeachclassismappedintoclassandmapsarehomotopicrelativetosubsetthatcontainsmultipointsclassesinducedmapsbetweenquotientspacesishomotopicrelativetoquotientofsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-325047329728180100Sun, 30 Aug 2026 14:19:32 +00002026-08-30T23:19:32.552+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for \(2\) continuous maps from same domain into same codomain and disjoint open cover of domain, if restrictions of maps on each element of cover are homotopic relative to subset, maps are homotopic relative to union of subsets Topics About: https://thebiasplanet.blogspot.com/2026/08/for2continuousmapsfromsamedomainintosamecodomainanddisjointopencoverofdomainifrestrictionsofmapsoneachelementofcoverarehomotopicrelativetosubsetmapsarehomotopicrelativetounionofsubsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8261828927507816991Sun, 30 Aug 2026 14:18:12 +00002026-08-30T23:18:12.969+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for \(2\) continuous maps from same domain into same codomain and finite disjoint closed cover of domain, if restrictions of maps on each element of cover are homotopic relative to subset, maps are homotopic relative to union of subsets Topics https://thebiasplanet.blogspot.com/2026/08/for2continuousmapsfromsamedomainintosamecodomainandfinitedisjointclosedcoverofdomainifrestrictionsofmapsoneachelementofcoverarehomotopicrelativetosubsetmapsarehomotopicrelativetounionofsubsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8336437566578997944Sun, 30 Aug 2026 14:16:59 +00002026-08-30T23:16:59.884+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that on set of continuous maps between topological spaces, for subset of domain, being homotopic relative to subset is equivalence relation Topics About: topological space The table of contents of this article Starting Context Target https://thebiasplanet.blogspot.com/2026/08/onsetofcontinuousmapsbetweentopologicalspacesforsubsetofdomainbeinghomotopicrelativetosubsetisequivalencerelation.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5524285306721015133Sun, 23 Aug 2026 13:25:35 +00002026-08-30T23:15:36.156+09:00Definitions and Propositions <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for continuous map from product of space and locally compact Hausdorff space and equivalence relations on 1st space and codomain, if 1st space equivalence class is mapped into codomain equivalence class, induced map between product of quotient space https://thebiasplanet.blogspot.com/2026/08/forcontinuousmapfromproductofspaceandlocallycompacthausdorffspaceandequivalencerelationson1stspaceandcodomainif1stspaceequivalenceclassismappedintocodomainequivalenceclassinducedmapbetweenproductofquotientspaceand2ndspaceandquotientspaceiscontinuous.htmlnoreply@blogger.com (Unknown)