tag:blogger.com,1999:blog-922111353248289998Wed, 27 May 2026 00:07:44 +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)Blogger2286125tag:blogger.com,1999:blog-922111353248289998.post-6704950754255913719Sun, 24 May 2026 11:54:45 +00002026-05-24T20:54:45.328+09:00Definitions and Propositions1799: For Sequence on \(1\)-Dimensional Euclidean Metric Space, if Sequence Converges, Sequence of Arithmetic Means of Leading Elements Converges with Convergence <The previous article in this series | The table of contents of this series | description/proof of that for sequence on \(1\)-dimensional Euclidean metric space, if sequence converges, sequence of arithmetic means of leading elements converges with convergence Topics About: metric space The table of contents of this article Starting Context Target Context Orientation https://thebiasplanet.blogspot.com/2026/05/forsequenceon1dimensionaleuclideanmetricspaceifsequenceconvergessequenceofarithmeticmeansofleadingelementsconvergeswithconvergence.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7663976925294553632Sun, 24 May 2026 11:53:05 +00002026-05-24T20:53:05.376+09:00Definitions and Propositions1798: For Non-Negative Measurable Map into \(1\)-Dimensional Euclidean Measurable Space, Composition of Floor Map After Map Is Measurable <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for non-negative measurable map into \(1\)-dimensional Euclidean measurable space, composition of floor Map after map is measurable Topics About: set The table of contents of this article Starting Context Target Context Orientation https://thebiasplanet.blogspot.com/2026/05/fornonnegativemeasurablemapinto1dimensionaleuclideanmeasurablespacecompositionoffloormapaftermapismeasurable.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1231405821031147551Sun, 24 May 2026 11:51:33 +00002026-05-24T20:51:33.829+09:00Definitions and Propositions1797: Floor Map with Codomain Extended to \(1\)-Dimensional Euclidean Measurable Space Is Measurable <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that floor map with codomain extended to \(1\)-dimensional Euclidean measurable space is measurable Topics About: measurable space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured https://thebiasplanet.blogspot.com/2026/05/floormapwithcodomainextendedto1dimensionaleuclideanmeasurablespaceismeasurable.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3341873966161633060Sun, 24 May 2026 11:50:04 +00002026-05-24T20:50:04.016+09:00Definitions and Propositions1796: Floor Map <The previous article in this series | The table of contents of this series | The next article in this series> definition of floor map 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 map. Target Context The https://thebiasplanet.blogspot.com/2026/05/floormap.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5591916700444196050Sun, 24 May 2026 11:48:28 +00002026-05-24T20:48:28.190+09:00Definitions and Propositions1795: Borel \(\sigma\)-Algebra of \(1\)-Dimensional Euclidean Topological Space Is Generated by Set of Upper-Open-or-Closed-Bounded Intervals, Set of Lower-Open-or-Closed-Bounded Intervals, or Set of Lower-Open-or-Closed-Upper-Open-or-Closed-Bounded Intervals <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that Borel \(\sigma\)-algebra of \(1\)-dimensional Euclidean topological space is generated by set of upper-open-or-closed-bounded intervals, set of lower-open-or-closed-bounded intervals, or set of lower-open-or-closed-upper-open-or-closed-bounded https://thebiasplanet.blogspot.com/2026/05/borelsigmaalgebraof1dimensionaleuclideantopologicalspaceisgeneratedbysetofupperopenorclosedboundedintervalssetofloweropenorclosedboundedintervalsorsetofloweropenorclosedupperopenorclosedboundedintervals.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7076987994814339647Sun, 24 May 2026 11:46:50 +00002026-05-24T20:46:50.061+09:00Definitions and Propositions1794: Half or Both Open Interval Is Union of Sequence of Closed Intervals <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that half or both open interval is union of sequence of closed intervals Topics About: set The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note 3: Proof Starting https://thebiasplanet.blogspot.com/2026/05/halforbothopenintervalisunionofsequenceofclosedintervals.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6285573383868875449Sun, 24 May 2026 11:45:19 +00002026-05-24T20:45:19.242+09:00Definitions and Propositions1793: Half or Both Closed Interval Is Intersection of Sequence of Open Intervals <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that half or both closed interval is intersection of sequence of open intervals 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/2026/05/halforbothclosedintervalisintersectionofsequenceofopenintervals.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3099799065767107603Sun, 24 May 2026 11:43:48 +00002026-05-24T20:43:48.725+09:00Definitions and Propositions1792: Series of \(1\)-Increasing Positive Natural Numbers to Powers of Minus Natural Number Larger than \(1\) Converges Smaller than This <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that series of \(1\)-increasing positive natural numbers to powers of minus natural number larger than \(1\) converges smaller than this Topics About: set The table of contents of this article Starting Context Target Context Orientation https://thebiasplanet.blogspot.com/2026/05/seriesof1increasingpositivenaturalnumberstopowersofminusnaturalnumberlargerthan1convergessmallerthanthis.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7455020355500845589Sun, 17 May 2026 12:45:52 +00002026-05-24T20:41:56.806+09:00Definitions and Propositions1791: For Topological Space That Is Union of Finite Number of Subspaces, Subset of Intersection of Subspaces That Is Open on Each Subspace Is Open on Base Space <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for topological space that is union of finite number of subspaces, subset of intersection of subspaces that is open on each subspace is open on base space Topics About: topological space The table of contents of this article Startinghttps://thebiasplanet.blogspot.com/2026/05/fortopologicalspacethatisunionoffinitenumberofsubspacessubsetofintersectionofsubspacesthatisopenoneachsubspaceisopenonbasespace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5157435576699864920Sun, 17 May 2026 12:44:02 +00002026-05-17T21:44:02.275+09:00Definitions and Propositions1790: For Probability Space, Independent Indexed Set of Events, and Indexed Set of Complements of Events, for Finite Subset of Index Set and Element of 1st Indexed Set or 2nd Indexed Set for Each Index, Measure of Intersection of Elements Is Product of Measures of 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 probability space, independent indexed set of events, and indexed set of complements of events, for finite subset of index set and element of 1st indexed set or 2nd indexed set for each index, measure of intersection of elements is product of https://thebiasplanet.blogspot.com/2026/05/forprobabilityspaceindependentindexedsetofeventsandindexedsetofcomplementsofeventsforfinitesubsetofindexsetandelementof1stindexedsetor2ndindexedsetforeachindexmeasureofintersectionofelementsisproductofmeasuresofelements.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4956452241452985824Sun, 17 May 2026 12:42:18 +00002026-05-17T21:42:18.307+09:00Definitions and Propositions1789: For Probability Space and Independent Indexed Set of Events, Indexed Set of Complements of Events Is Independent <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 and independent indexed set of events, indexed set of complements of events is independent Topics About: measure space The table of contents of this article Starting Context Target Context Orientation Main https://thebiasplanet.blogspot.com/2026/05/forprobabilityspaceandindependentindexedsetofeventsindexedsetofcomplementsofeventsisindependent.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-339798385139386015Sun, 17 May 2026 12:40:50 +00002026-05-17T21:40:50.170+09:00Definitions and Propositions1788: For Probability Space and Independent Indexed Set of Events, for Finite Indexed Subset of Indexed Set, \(1\) Minus Probability of Union of Indexed Subset Is Product of \(1\) Minus Probabilities of Elements of Indexed Subset <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 and independent indexed set of events, for finite indexed subset of indexed set, \(1\) minus probability of union of indexed subset is product of \(1\) minus probabilities of elements of indexed subset Topics About: https://thebiasplanet.blogspot.com/2026/05/forprobabilityspaceandindependentindexedsetofeventsforfiniteindexedsubsetofindexedset1minusprobabilityofunionofindexedsubsetisproductof1minusprobabilitiesofelementsofindexedsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6987353354865006014Sun, 17 May 2026 12:39:22 +00002026-05-17T21:39:22.251+09:00Definitions and Propositions1787: For Probability Space and Independent Indexed Set of Events, Indexed Set of Events by Taking Union of Some Finite Elements Is Independent <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 and independent indexed set of events, indexed set of events by taking union of some finite elements is independent Topics About: measure space The table of contents of this article Starting Context Target https://thebiasplanet.blogspot.com/2026/05/forprobabilityspaceandindependentindexedsetofeventsindexedsetofeventsbytakingunionofsomefiniteelementsisindependent.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3900094943952267744Sun, 17 May 2026 12:37:42 +00002026-05-17T21:37:42.249+09:00Definitions and Propositions1786: Independent Indexed Set of Events of Probability Space <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 events of probability space Topics About: measure space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description Starting Context The reader knows a https://thebiasplanet.blogspot.com/2026/05/independentindexedsetofeventsofprobabilityspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7254078196778552962Sun, 17 May 2026 12:36:03 +00002026-05-17T21:36:03.967+09:00Definitions and Propositions1785: Probability Space <The previous article in this series | The table of contents of this series | The next article in this series> definition 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 The reader knows a definition of measure space. https://thebiasplanet.blogspot.com/2026/05/probabilityspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-306087190348976335Sun, 17 May 2026 12:34:24 +00002026-05-17T21:34:24.068+09:00Definitions and Propositions1784: For Measure Space and \(2\) Measurable Subsets, Measure of Union of Subsets Is Sum of Measures of Subsets Minus Measure of Intersection of Subsets <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for measure space and \(2\) measurable subsets, measure of union of subsets is sum of measures of subsets minus measure of intersection of subsets Topics About: measure space The table of contents of this article Starting Context https://thebiasplanet.blogspot.com/2026/05/formeasurespaceand2measurablesubsetsmeasureofunionofsubsetsissumofmeasuresofsubsetsminusmeasureofintersectionofsubsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3876236323358331218Sun, 17 May 2026 12:32:53 +00002026-05-17T21:32:53.113+09:00Definitions and Propositions1783: For Map and Subset of Codomain, Complement of Preimage of Subset Is Preimage of Complement of Subset <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for map and subset of codomain, complement of preimage of subset is preimage of complement of subset Topics About: set The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured https://thebiasplanet.blogspot.com/2026/05/formapandsubsetofcodomaincomplementofpreimageofsubsetispreimageofcomplementofsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8654500142722788239Sun, 17 May 2026 12:31:14 +00002026-05-17T21:31:14.867+09:00Definitions and Propositions1782: For Continuous Map from Closed Subset of Normal Topological Space into Interval, There Is Continuous Extension of Map into Same Interval (Tietze Extension Theorem) <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 closed subset of normal topological space into interval, there is continuous extension of map into same interval (Tietze extension theorem) Topics About: topological space The table of contents of this article https://thebiasplanet.blogspot.com/2026/05/forcontinuousmapfromclosedsubsetofnormaltopologicalspaceintointervalthereiscontinuousextensionofmapintosameintervaltietzeextensiontheorem.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7167871230468152073Sun, 17 May 2026 12:29:43 +00002026-05-17T21:29:43.818+09:00Definitions and Propositions1781: For Continuous Map from Closed Subset of Topological Space and Closed Subset and Open Subset Containing Closed Subset of Codomain, There Is Open Subset of Domain Space That Contains Preimage of Closed Subset s.t. Complement of Open Subset Contains Preimage of Complement of Open Subset <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 closed subset of topological space and closed subset and open subset containing closed subset of codomain, there is open subset of domain space that contains preimage of closed subset s.t. complement of open subset contains https://thebiasplanet.blogspot.com/2026/05/forcontinuousmapfromclosedsubsetoftopologicalspaceandclosedsubsetandopensubsetcontainingclosedsubsetofcodomainthereisopensubsetofdomainspacethatcontainspreimageofclosedsubsetstcomplementofopensubsetcontainspreimageofcomplementofopensubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5444986298731308386Sun, 17 May 2026 12:28:02 +00002026-05-17T21:28:02.519+09:00Definitions and Propositions1780: For \(2\) Sets, Maps from 1st Set into 2nd Set Are Same iff Preimages of Each Subset of Codomain Are Same <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for \(2\) sets, maps from 1st set into 2nd set are same iff preimages of each subset of codomain are same Topics About: set The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured https://thebiasplanet.blogspot.com/2026/05/for2setsmapsfrom1stsetinto2ndsetaresameiffpreimagesofeachsubsetofcodomainaresame.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-781933864174843752Sun, 17 May 2026 12:26:20 +00002026-05-17T21:26:20.157+09:00Definitions and Propositions1779: For \(2\) Sets, Union of Sets Is Exclusive Union of 1st Set Minus Intersection of Sets, 2nd Set Minus Intersection of Sets, and Intersection of Sets <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for \(2\) sets, union of sets is exclusive union of 1st set minus intersection of sets, 2nd set minus intersection of sets, and intersection of sets Topics About: set The table of contents of this article Starting Context Target https://thebiasplanet.blogspot.com/2026/05/for2setsunionofsetsisexclusiveunionof1stsetminusintersectionofsets2ndsetminusintersectionofsetsandintersectionofsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-620619385345619807Sun, 17 May 2026 12:24:32 +00002026-05-17T21:24:32.131+09:00Definitions and Propositions1778: Indexed Set <The previous article in this series | The table of contents of this series | The next article in this series> definition of indexed set 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 map. Target Context The https://thebiasplanet.blogspot.com/2026/05/indexedset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5735699394296604056Sun, 10 May 2026 15:02:00 +00002026-05-17T21:22:32.610+09:00Definitions and Propositions1777: Adjunction Topological Space Is Path-Connected if Domain of Attaching Map Is Nonempty and Attaching-Origin Space and Attaching-Destination Space Are 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 adjunction topological space is path-connected if domain of attaching map is nonempty and attaching-origin space and attaching-destination space are path-connected Topics About: topological space The table of contents of this article https://thebiasplanet.blogspot.com/2026/05/adjunctiontopologicalspaceispathconnectedifdomainofattachingmapisnonemptyandattachingoriginspaceandattachingdestinationspacearepathconnected.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8887715485883946124Sun, 10 May 2026 15:00:00 +00002026-05-11T00:00:27.396+09:00Definitions and Propositions1776: Adjunction Topological Space Is Connected if Domain of Attaching Map Is Nonempty and Attaching-Origin Space and Attaching-Destination Space Are Connected <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that adjunction topological space is connected if domain of attaching map is nonempty and attaching-origin space and attaching-destination space are connected Topics About: topological space The table of contents of this article Starting https://thebiasplanet.blogspot.com/2026/05/adjunctiontopologicalspaceisconnectedifdomainofattachingmapisnonemptyandattachingoriginspaceandattachingdestinationspaceareconnected.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4131138828886837519Sun, 10 May 2026 14:58:00 +00002026-05-10T23:58:37.678+09:00Definitions and Propositions1775: For Topological Sum, Constituent Is Topological Subspace of Topological Sum <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for topological sum, constituent is topological subspace of topological sum Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: https://thebiasplanet.blogspot.com/2026/05/fortopologicalsumconstituentistopologicalsubspaceoftopologicalsum.htmlnoreply@blogger.com (Unknown)