tag:blogger.com,1999:blog-922111353248289998Wed, 20 May 2026 00:37:28 +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)Blogger2278125tag:blogger.com,1999:blog-922111353248289998.post-7455020355500845589Sun, 17 May 2026 12:45:52 +00002026-05-17T21:45:52.162+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 | 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 Starting Context Target Context Orientationhttps://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)tag:blogger.com,1999:blog-922111353248289998.post-8526095350586071108Sun, 10 May 2026 14:57:00 +00002026-05-10T23:57:08.071+09:00Definitions and Propositions1774: Normal Topological Space Is Completely Regular <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that normal topological space is completely regular 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 The https://thebiasplanet.blogspot.com/2026/05/normaltopologicalspaceiscompletelyregular.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8864447345940011840Sun, 10 May 2026 14:55:00 +00002026-05-10T23:55:32.547+09:00Definitions and Propositions1773: Normal Topological Space Is Regular <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that normal topological space is regular 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 The reader knowshttps://thebiasplanet.blogspot.com/2026/05/normaltopologicalspaceisregular.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2525699700743845076Sun, 10 May 2026 14:53:00 +00002026-05-10T23:53:52.044+09:00Definitions and Propositions1772: Completely Regular Topological Space Is Regular <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that completely regular topological space is regular 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 The https://thebiasplanet.blogspot.com/2026/05/completelyregulartopologicalspaceisregular.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-412354262330765553Sun, 10 May 2026 14:52:00 +00002026-05-10T23:52:13.843+09:00Definitions and Propositions1771: Regular Topological Space Is Hausdorff <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that regular topological space is Hausdorff 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 The reader https://thebiasplanet.blogspot.com/2026/05/regulartopologicalspaceishausdorff.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7118446626337766698Sun, 10 May 2026 14:50:00 +00002026-05-10T23:50:34.519+09:00Definitions and Propositions1770: Trivial Topological Space <The previous article in this series | The table of contents of this series | The next article in this series> definition of trivial topological space 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 The reader knows a definition of https://thebiasplanet.blogspot.com/2026/05/trivialtopologicalspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2213639705801324371Sun, 10 May 2026 14:48:00 +00002026-05-10T23:48:37.542+09:00Definitions and Propositions1769: For Normal Topological Space, Closed Subset, and Open Subset That Contains Closed Subset, There Is Continuous Map into Closed Interval That Maps Closed Subset to Boundary and Complement of Open Subset to Other Boundary (Urysohn's Lemma) <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for normal topological space, closed subset, and open subset that contains closed subset, there is continuous map into closed Interval that maps closed subset to boundary and complement of open subset to other boundary (Urysohn's lemma) Topics https://thebiasplanet.blogspot.com/2026/05/fornormaltopologicalspaceclosedsubsetandopensubsetthatcontainsclosedsubsetthereiscontinuousmapintoclosedintervalthatmapsclosedsubsettoboundaryandcomplementofopensubsettootherboundaryurysohnslemma.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4594939733679603304Sun, 10 May 2026 14:46:00 +00002026-05-10T23:46:56.520+09:00Definitions and Propositions1768: For Metric Space and Subset, Subset Is Totally Bounded iff Closure of Subset Is Totally Bounded iff Closure of Subset Is Totally Bounded with Centers in 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 metric space and subset, subset is totally bounded iff closure of subset is totally bounded iff closure of subset is totally bounded with centers in subset Topics About: metric space About: topological space The table https://thebiasplanet.blogspot.com/2026/05/formetricspaceandsubsetsubsetistotallyboundediffclosureofsubsetistotallyboundediffclosureofsubsetistotallyboundedwithcentersinsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5895490153985891445Mon, 04 May 2026 11:45:00 +00002026-05-10T23:45:23.256+09:00Definitions and Propositions1767: Finite Product of Topological Sums Is Topological Sum of Products <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that finite product of topological sums is topological sum of products Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Proof https://thebiasplanet.blogspot.com/2026/05/finiteproductoftopologicalsumsistopologicalsumofproducts.htmlnoreply@blogger.com (Unknown)