tag:blogger.com,1999:blog-922111353248289998Sun, 23 Aug 2026 13:25:35 +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 CommandsT.B.P.https://thebiasplanet.blogspot.com/noreply@blogger.com (Unknown)Blogger2439125tag:blogger.com,1999:blog-922111353248289998.post-5524285306721015133Sun, 23 Aug 2026 13:25:35 +00002026-08-23T22:25:35.156+09:00Definitions and Propositions1952: 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 and 2nd Space and Quotient Space Is Continuous <The previous article in this series | The table of contents of 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 and 2nd space and quotient space is https://thebiasplanet.blogspot.com/2026/08/forcontinuousmapfromproductofspaceandlocallycompacthausdorffspaceandequivalencerelationson1stspaceandcodomainif1stspaceequivalenceclassismappedintocodomainequivalenceclassinducedmapbetweenproductofquotientspaceand2ndspaceandquotientspaceiscontinuous.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-721493332387362634Sun, 23 Aug 2026 13:24:10 +00002026-08-23T22:24:10.697+09:00Definitions and Propositions1951: For Continuous Map and Equivalence Relations on Domain and Codomain, if Domain Equivalence Class Is Mapped into Codomain Equivalence Class, Induced Map Between Quotient Spaces Is Continuous <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 and equivalence relations on domain and codomain, if domain equivalence class is mapped into codomain equivalence class, induced map between quotient spaces is continuous Topics About: topological space The table ofhttps://thebiasplanet.blogspot.com/2026/08/forcontinuousmapandequivalencerelationsondomainandcodomainifdomainequivalenceclassismappedintocodomainequivalenceclassinducedmapbetweenquotientspacesiscontinuous.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2581937443540433048Sun, 23 Aug 2026 13:22:44 +00002026-08-23T22:22:44.847+09:00Definitions and Propositions1950: For Contractible Topological Space, Each Constant Topological Space Endomorphism Is Homotopic to Identity Map <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for contractible topological space, each constant topological space endomorphism is homotopic to identity map Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Bodyhttps://thebiasplanet.blogspot.com/2026/08/forcontractibletopologicalspaceeachconstanttopologicalspaceendomorphismishomotopictoidentitymap.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5610554354074396292Sun, 23 Aug 2026 13:21:24 +00002026-08-23T22:21:24.467+09:00Definitions and Propositions1949: \(2\) Continuous Maps from Contractible Topological Space into Path-Connected Topological Space Are Homotopic <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that \(2\) continuous maps from contractible topological space into path-connected topological space are homotopic Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Bodyhttps://thebiasplanet.blogspot.com/2026/08/2continuousmapsfromcontractibletopologicalspaceintopathconnectedtopologicalspacearehomotopic.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8161851333353146331Sun, 23 Aug 2026 13:20:01 +00002026-08-23T22:20:01.987+09:00Definitions and Propositions1948: \(2\) Constant Maps from Topological Space into Path-Connected Topological Space Are Homotopic <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that \(2\) constant maps from topological space into path-connected topological space are homotopic Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured https://thebiasplanet.blogspot.com/2026/08/2constantmapsfromtopologicalspaceintopathconnectedtopologicalspacearehomotopic.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3783058043780999169Sun, 23 Aug 2026 13:18:20 +00002026-08-23T22:18:20.759+09:00Definitions and Propositions1947: Continuous Map from Contractible Topological Space into Topological Space Is Homotopic to Constant Map <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that continuous map from contractible topological space into topological space is homotopic to constant map Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: https://thebiasplanet.blogspot.com/2026/08/continuousmapfromcontractibletopologicalspaceintotopologicalspaceishomotopictoconstantmap.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3907660686557014672Sun, 23 Aug 2026 13:17:02 +00002026-08-23T22:17:02.691+09:00Definitions and Propositions1946: Contractible Topological Space Is Path-Connected <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that contractible topological space is path-connected Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Proof Starting Context https://thebiasplanet.blogspot.com/2026/08/contractibletopologicalspaceispathconnected.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6404277184502557394Sun, 23 Aug 2026 13:15:37 +00002026-08-23T22:15:37.332+09:00Definitions and Propositions1945: Topological Space Is Contractible iff Space Is Homotopy Equivalent to \(1\)-Point Topological Space <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that topological space is contractible iff space is homotopy equivalent to \(1\)-point topological space Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: https://thebiasplanet.blogspot.com/2026/08/topologicalspaceiscontractibleiffspaceishomotopyequivalentto1pointtopologicalspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2420020088313159051Sun, 23 Aug 2026 13:14:02 +00002026-08-23T22:14:02.314+09:00Definitions and Propositions1944: Homotopy Equivalence Relation on Collection of Topological Spaces <The previous article in this series | The table of contents of this series | The next article in this series> definition of homotopy equivalence relation on collection of topological spaces Topics About: topological 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/2026/08/homotopyequivalencerelationoncollectionoftopologicalspaces.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3125939343817579961Sun, 23 Aug 2026 13:12:33 +00002026-08-23T22:12:33.905+09:00Definitions and Propositions1943: Quotient Collection <The previous article in this series | The table of contents of this series | The next article in this series> definition of quotient collection Topics About: collection 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 equivalence https://thebiasplanet.blogspot.com/2026/08/quotientcollection.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1716341159168452280Sun, 23 Aug 2026 13:11:07 +00002026-08-23T22:11:07.056+09:00Definitions and Propositions1942: Equivalence Relation on Collection <The previous article in this series | The table of contents of this series | The next article in this series> definition of equivalence relation on collection Topics About: collection 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/2026/08/equivalencerelationoncollection.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3334967543687684725Sun, 16 Aug 2026 13:38:45 +00002026-08-23T22:09:41.461+09:00Definitions and Propositions1941: For Linearly-Ordered Set, Intersection of \(2\) Intervals Is Interval as This <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for linearly-ordered set, intersection of \(2\) intervals is interval as this Topics About: set 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/forlinearlyorderedsetintersectionof2intervalsisintervalasthis.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5674304124444807352Sun, 16 Aug 2026 13:37:18 +00002026-08-16T22:37:18.196+09:00Definitions and Propositions1940: For Linearly-Ordered Set and \(2\) Elements, Element Is Smaller than 1st Element and Is Equal to or Smaller than 2nd Element iff This <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for linearly-ordered set and \(2\) elements, element is smaller than 1st element and is equal to or smaller than 2nd element iff this Topics About: set The table of contents of this article Starting Context Target Context https://thebiasplanet.blogspot.com/2026/08/forlinearlyorderedsetand2elementselementissmallerthan1stelementandisequaltoorsmallerthan2ndelementiffthis.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1936424887245071970Sun, 16 Aug 2026 13:35:37 +00002026-08-16T22:35:37.203+09:00Definitions and Propositions1939: For Linearly-Ordered Set and \(2\) Elements, Element Is Larger than 1st Element and Is Equal to or Larger than 2nd Element iff This <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for linearly-ordered set and \(2\) elements, element is larger than 1st element and is equal to or larger than 2nd element iff this Topics About: set The table of contents of this article Starting Context Target Context Orientation https://thebiasplanet.blogspot.com/2026/08/forlinearlyorderedsetand2elementselementislargerthan1stelementandisequaltoorlargerthan2ndelementiffthis.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5306431051691493753Sun, 16 Aug 2026 13:34:10 +00002026-08-16T22:34:10.142+09:00Definitions and Propositions1938: For Linearly-Ordered Set and \(2\) Elements, Element Is Equal to or Smaller than 1st Element and Is Equal to or Smaller than 2nd Element iff Element Is Equal to or Smaller than Minimum of \(2\) Elements <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for linearly-ordered set and \(2\) elements, element is equal to or smaller than 1st element and is equal to or smaller than 2nd element iff element is equal to or smaller than minimum of \(2\) elements Topics About: set The table ofhttps://thebiasplanet.blogspot.com/2026/08/forlinearlyorderedsetand2elementselementisequaltoorsmallerthan1stelementandisequaltoorsmallerthan2ndelementiffelementisequaltoorsmallerthanminimumof2elements.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1834485994565939179Sun, 16 Aug 2026 13:32:37 +00002026-08-16T22:32:37.431+09:00Definitions and Propositions1937: For Linearly-Ordered Set and \(2\) Elements, Element Is Smaller than 1st Element and Is Smaller than 2nd Element iff Element Is Smaller than Minimum of \(2\) Elements <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for linearly-ordered set and \(2\) elements, element is smaller than 1st element and is smaller than 2nd element iff element is smaller than minimum of \(2\) elements Topics About: set The table of contents of this article Starting https://thebiasplanet.blogspot.com/2026/08/forlinearlyorderedsetand2elementselementissmallerthan1stelementandissmallerthan2ndelementiffelementissmallerthanminimumof2elements.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2108286181463379368Sun, 16 Aug 2026 13:31:04 +00002026-08-16T22:31:04.770+09:00Definitions and Propositions1936: For Linearly-Ordered Set and \(2\) Elements, Element Is Equal to or Larger than 1st Element and Is Equal to or Larger than 2nd Element iff Element Is Equal to or Larger than Maximum of \(2\) Elements <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for linearly-ordered set and \(2\) elements, element is equal to larger than 1st element and is equal to larger than 2nd element iff element is equal to larger than maximum of \(2\) elements Topics About: set The table of contents ofhttps://thebiasplanet.blogspot.com/2026/08/forlinearlyorderedsetand2elementselementisequaltoorlargerthan1stelementandisequaltoorlargerthan2ndelementiffelementisequaltoorlargerthanmaximumof2elements.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8814529571664229779Sun, 16 Aug 2026 13:29:33 +00002026-08-16T22:29:33.277+09:00Definitions and Propositions1935: For Linearly-Ordered Set and \(2\) Elements, Element Is Larger than 1st Element and Is Larger than 2nd Element iff Element Is Larger than Maximum of \(2\) Elements <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for linearly-ordered set and \(2\) elements, element is larger than 1st element and is larger than 2nd element iff element is larger than maximum of \(2\) elements Topics About: set The table of contents of this article Starting https://thebiasplanet.blogspot.com/2026/08/forlinearlyorderedsetand2elementselementislargerthan1stelementandislargerthan2ndelementiffelementislargerthanmaximumof2elements.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4034216491376887836Sun, 16 Aug 2026 13:28:07 +00002026-08-16T22:28:07.964+09:00Definitions and Propositions1934: For Real Number, There Is Decreasing Sequence with Rational Values Possibly with Denominators with Same Base Which Converges to Real Number <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for real number, there is decreasing sequence with rational values possibly with denominators with same base which converges to real number Topics About: metric space The table of contents of this article Starting Context Target https://thebiasplanet.blogspot.com/2026/08/forrealnumberthereisdecreasingsequencewithrationalvaluespossiblywithdenominatorswithsamebasewhichconvergestorealnumber.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3162159876585707269Sun, 16 Aug 2026 13:26:34 +00002026-08-16T22:26:34.729+09:00Definitions and Propositions1933: For Real Number, There Is Increasing Sequence with Rational Values Possibly with Denominators with Same Base Which Converges to Real Number <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for real number, there is increasing sequence with rational values possibly with denominators with same base which converges to real number Topics About: metric space The table of contents of this article Starting Context Target https://thebiasplanet.blogspot.com/2026/08/forrealnumberthereisincreasingsequencewithrationalvaluespossiblywithdenominatorswithsamebasewhichconvergestorealnumber.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7274975274906108545Sun, 16 Aug 2026 13:25:14 +00002026-08-16T22:25:14.204+09:00Definitions and Propositions1932: Quadratic Form by \(2\)-Symmetric-Tensor Determines Tensor <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that quadratic form by \(2\)-symmetric-tensor determines tensor Topics About: vectors space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note 3: Proof Startinghttps://thebiasplanet.blogspot.com/2026/08/quadraticformby2symmetrictensordeterminestensor.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-196967273631883823Sun, 16 Aug 2026 13:23:21 +00002026-08-16T22:23:21.729+09:00Definitions and Propositions1931: Quadratic Form by \(2\)-Symmetric-Tensor <The previous article in this series | The table of contents of this series | The next article in this series> definition of quadratic form by \(2\)-symmetric-tensor Topics About: vectors space 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 https://thebiasplanet.blogspot.com/2026/08/quadraticformby2symmetrictensor.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2983618318382236309Sun, 16 Aug 2026 13:21:56 +00002026-08-16T22:21:56.081+09:00Definitions and Propositions1930: Union of Subsequences of Sequence <The previous article in this series | The table of contents of this series | The next article in this series> definition of union of subsequences of sequence 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 subsequence https://thebiasplanet.blogspot.com/2026/08/unionofsubsequencesofsequence.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2441563296460022625Sun, 16 Aug 2026 13:20:32 +00002026-08-16T22:20:32.678+09:00Definitions and Propositions1929: Canonical Subsequence for Subsequence of Sequence <The previous article in this series | The table of contents of this series | The next article in this series> definition of canonical subsequence for subsequence of sequence 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 definitionhttps://thebiasplanet.blogspot.com/2026/08/canonicalsubsequenceforsubsequenceofsequence.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7078559374823125910Sun, 09 Aug 2026 12:48:43 +00002026-08-16T22:19:04.511+09:00Definitions and Propositions1928: For Infinite Sequence on Partially-Ordered Set and Element of Set, if There Is Any Large Index Whose Value Is Equal to or Larger than Element, Limit Superior Is Equal to or Larger than Element <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for infinite sequence on partially-ordered set and element of set, if there is any large index whose value is equal to or larger than element, limit superior is equal to or larger than element Topics About: set The table of contents https://thebiasplanet.blogspot.com/2026/08/forinfinitesequenceonpartiallyorderedsetandelementofsetifthereisanylargeindexwhosevalueisequaltoorlargerthanelementlimitsuperiorisequaltoorlargerthanelement.htmlnoreply@blogger.com (Unknown)