tag:blogger.com,1999:blog-922111353248289998Sun, 02 Aug 2026 13:18:51 +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)Blogger2403125tag:blogger.com,1999:blog-922111353248289998.post-1172781633145857234Sun, 02 Aug 2026 13:18:51 +00002026-08-02T22:18:51.326+09:00Definitions and Propositions1916: \(\sigma\)-Algebra Induced on Domain of Maps into Measurable Space <The previous article in this series | The table of contents of this series | definition of \(\sigma\)-algebra induced on domain of maps into measurable space Topics About: measurable 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/2026/08/sigmaalgebrainducedondomainofmapsintomeasurablespace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3854735008002065632Sun, 02 Aug 2026 13:17:26 +00002026-08-02T22:17:26.038+09:00Definitions and Propositions1915: For Set and Sets of Subsets, \(\sigma\)-Algebra Generated by Union of \(\sigma\)-Algebras Generated by Sets of Subsets Is \(\sigma\)-Algebra Generated by Union of Sets 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 set and sets of subsets, \(\sigma\)-algebra generated by union of \(\sigma\)-algebras generated by sets of subsets is \(\sigma\)-algebra generated by union of sets of subsets Topics About: measurable space The table of contents https://thebiasplanet.blogspot.com/2026/08/forsetandsetsofsubsetssigmaalgebrageneratedbyunionofsigmaalgebrasgeneratedbysetsofsubsetsissigmaalgebrageneratedbyunionofsetsofsubsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6402860080527790756Sun, 02 Aug 2026 13:15:50 +00002026-08-02T22:15:50.621+09:00Definitions and Propositions1914: For Set and Set of Subsets, \(\sigma\)-Algebra Generated by \(\sigma\)-Algebra Generated by Set of Subsets Is \(\sigma\)-Algebra Generated by Set of Subsets <The previous article in this series | The table of contents of this series | The next article in this series> 1914: For Set and Set of Subsets, \(\sigma\)-Algebra Generated by \(\sigma\)-Algebra Generated by Set of Subsets Is \(\sigma\)-Algebra Generated by Set of Subsets Topics About: measurable space The table of contents of this article Starting Context Target https://thebiasplanet.blogspot.com/2026/08/forsetandsetofsubsetssigmaalgebrageneratedbysigmaalgebrageneratedbysetofsubsetsissigmaalgebrageneratedbysetofsubsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5825213468446657864Sun, 02 Aug 2026 13:14:16 +00002026-08-02T22:14:16.730+09:00Definitions and Propositions1913: For Set and \(2\) Sets of Subsets, if Former Set of Subsets Is Contained in Latter Set of Subsets, \(\sigma\)-Algebra Generated by Former Set Is Contained in \(\sigma\)-Algebra Generated by Latter Set <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for set and \(2\) sets of subsets, if former set of subsets is contained in latter set of subsets, \(\sigma\)-algebra generated by former set is contained in \(\sigma\)-algebra generated by latter set Topics About: measurable space https://thebiasplanet.blogspot.com/2026/08/forsetand2setsofsubsetsifformersetofsubsetsiscontainedinlattersetofsubsetssigmaalgebrageneratedbyformersetiscontainedinsigmaalgebrageneratedbylatterset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4305719514368605657Sun, 02 Aug 2026 13:12:53 +00002026-08-02T22:12:53.960+09:00Definitions and Propositions1912: \(\sigma\)-Algebra Induced on Domain of Map into Measurable Space <The previous article in this series | The table of contents of this series | The next article in this series> definition of \(\sigma\)-algebra induced on domain of map into measurable space Topics About: measurable 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/sigmaalgebrainducedondomainofmapintomeasurablespace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5345119632669430842Sun, 02 Aug 2026 13:11:29 +00002026-08-02T22:11:29.229+09:00Definitions and Propositions1911: Euclidean Measurable Space <The previous article in this series | The table of contents of this series | The next article in this series> definition of Euclidean measurable space Topics About: measurable 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/2026/08/euclideanmeasurablespace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8558540349867254760Sun, 02 Aug 2026 13:10:07 +00002026-08-02T22:10:07.099+09:00Definitions and Propositions1910: For Non-Decreasing Sequence on \(1\)-Dimensional Euclidean Metric Space with Canonical Ordering, Convergence of Sequence Is Supremum of Range of Sequence <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-decreasing sequence on \(1\)-dimensional Euclidean metric space with canonical ordering, convergence of sequence is supremum of range of sequence Topics About: metric space The table of contents of this article Starting https://thebiasplanet.blogspot.com/2026/08/fornondecreasingsequenceon1dimensionaleuclideanmetricspacewithcanonicalorderingconvergenceofsequenceissupremumofrangeofsequence.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8526345020178504482Sun, 02 Aug 2026 13:08:40 +00002026-08-02T22:08:40.606+09:00Definitions and Propositions1909: For Non-Increasing Sequence on \(1\)-Dimensional Euclidean Metric Space with Canonical Ordering, Convergence of Sequence Is Infimum of Range of Sequence <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-increasing sequence on \(1\)-dimensional Euclidean metric space with canonical ordering, convergence of sequence is infimum of range of sequence Topics About: metric space The table of contents of this article Starting https://thebiasplanet.blogspot.com/2026/08/fornonincreasingsequenceon1dimensionaleuclideanmetricspacewithcanonicalorderingconvergenceofsequenceisinfimumofrangeofsequence.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1884606421264282815Sun, 02 Aug 2026 13:07:17 +00002026-08-02T22:07:17.223+09:00Definitions and Propositions1908: Finite Product of Open Quotient Maps Is Open Quotient <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 open quotient maps is open quotient 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 Starting https://thebiasplanet.blogspot.com/2026/08/finiteproductofopenquotientmapsisopenquotient.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4716933510955288271Sun, 02 Aug 2026 13:05:50 +00002026-08-02T22:05:50.930+09:00Definitions and Propositions1907: For Quotient Map from 1st Topological Space onto 2nd Topological Space and Identity Map over 3rd Locally Compact Hausdorff Topological Space, Product of Map and Identity Map Is Quotient <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for quotient map from 1st topological space onto 2nd topological space and identity map over 3rd locally compact Hausdorff topological space, product of map and identity map is quotient Topics About: topological space The table of https://thebiasplanet.blogspot.com/2026/08/forquotientmapfrom1sttopologicalspaceonto2ndtopologicalspaceandidentitymapover3rdlocallycompacthausdorfftopologicalspaceproductofmapandidentitymapisquotient.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8617912268114052492Sun, 02 Aug 2026 13:04:12 +00002026-08-02T22:04:12.231+09:00Definitions and Propositions1906: For Map from Product of \(2\) Sets into Set, Subset of 2nd Set, and Subset of 3rd Set, Complement of Subset of 1st s.t. Image of Product of Point and Subset of 2nd Is Contained in Subset of 3rd Is Projection of Intersection of Preimage of Complement of Subset of 3rd and Product of 1st and Subset of 2nd <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 from product of \(2\) sets into set, subset of 2nd set, and subset of 3rd set, complement of subset of 1st s.t. image of product of point and subset of 2nd is contained in subset of 3rd is projection of intersection of preimage of complement https://thebiasplanet.blogspot.com/2026/08/formapfromproductof2setsintosetsubsetof2ndsetandsubsetof3rdsetcomplementofsubsetof1ststimageofproductofpointandsubsetof2ndiscontainedinsubsetof3rdisprojectionofintersectionofpreimageofcomplementofsubsetof3rdandproductof1standsubsetof2nd.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2738824854062460289Sun, 02 Aug 2026 13:02:43 +00002026-08-02T22:02:43.645+09:00Definitions and Propositions1905: For Set with Equivalence Relation, Subset, Canonical Injection from Quotient Set into Quotient Set, and Subset of Quotient Set of Subset, Preimage Under Classification Map of Image of Subset Under Canonical Injection Is Saturation of Preimage Under Classification Map 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 set with equivalence relation, subset, canonical injection from quotient set into quotient set, and subset of quotient set of subset, preimage under classification map of image of subset under canonical injection is saturation of preimage under https://thebiasplanet.blogspot.com/2026/08/forsetwithequivalencerelationsubsetcanonicalinjectionfromquotientsetintoquotientsetandsubsetofquotientsetofsubsetpreimageunderclassificationmapofimageofsubsetundercanonicalinjectionissaturationofpreimageunderclassificationmapofsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6470594760193094709Sun, 02 Aug 2026 13:01:13 +00002026-08-02T22:01:13.009+09:00Definitions and Propositions1904: Saturation of Subset of Set with Equivalence Relation Is Set iff Canonical Injection from Quotient Set of Subset into Quotient Set of Set Is Bijection <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that saturation of subset of set with equivalence relation is set iff canonical injection from quotient set of subset into quotient set of set is bijection Topics About: set The table of contents of this article Starting Context Target https://thebiasplanet.blogspot.com/2026/08/saturationofsubsetofsetwithequivalencerelationissetiffcanonicalinjectionfromquotientsetofsubsetintoquotientsetofsetisbijection.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1988771392132931872Sun, 02 Aug 2026 12:59:49 +00002026-08-02T21:59:49.348+09:00Definitions and Propositions1903: Saturation of Subset of Set with Equivalence Relation Is Subset iff Subset Is Union of Equivalence Classes <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that saturation of subset of set with equivalence relation is subset iff subset is union of equivalence classes Topics About: set The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured https://thebiasplanet.blogspot.com/2026/08/saturationofsubsetofsetwithequivalencerelationissubsetiffsubsetisunionofequivalenceclasses.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8248816166321228379Sun, 02 Aug 2026 12:58:22 +00002026-08-02T21:58:22.049+09:00Definitions and Propositions1902: Saturation of Subset of Set with Equivalence Relation <The previous article in this series | The table of contents of this series | The next article in this series> definition of saturation of subset of set with equivalence relation 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 https://thebiasplanet.blogspot.com/2026/08/saturationofsubsetofsetwithequivalencerelation.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6495382566422919465Sun, 26 Jul 2026 12:58:18 +00002026-08-02T21:56:54.842+09:00Definitions and Propositions1901: For Topological Space with Equivalence Relation and Subspace with Subset Equivalence Relation, Canonical Injection from Quotient Topological Space of Subspace into Quotient Topological Space of Space 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 topological space with equivalence relation and subspace with subset equivalence relation, canonical injection from quotient topological space of subspace into quotient topological space of space is continuous Topics About: topological https://thebiasplanet.blogspot.com/2026/07/fortopologicalspacewithequivalencerelationandsubspacewithsubsetequivalencerelationcanonicalinjectionfromquotienttopologicalspaceofsubspaceintoquotienttopologicalspaceofspaceiscontinuous.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-502043548067559874Sun, 26 Jul 2026 12:56:55 +00002026-07-26T21:56:55.767+09:00Definitions and Propositions1900: For Set with Equivalence Relation and Subset with Subset Equivalence Relation, There Is Canonical Injection from Quotient Set of Subset into Quotient Set of Set <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for set with equivalence relation and subset with subset equivalence relation, there is canonical injection from quotient set of subset into quotient set of set Topics About: set The table of contents of this article Starting https://thebiasplanet.blogspot.com/2026/07/forsetwithequivalencerelationandsubsetwithsubsetequivalencerelationthereiscanonicalinjectionfromquotientsetofsubsetintoquotientsetofset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8366389928301501024Sun, 26 Jul 2026 12:55:24 +00002026-07-26T21:55:24.645+09:00Definitions and Propositions1899: Subset Equivalence Relation <The previous article in this series | The table of contents of this series | The next article in this series> definition of subset equivalence relation 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 equivalence https://thebiasplanet.blogspot.com/2026/07/subsetequivalencerelation.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7445853780278108494Sun, 26 Jul 2026 12:53:27 +00002026-07-26T21:53:27.705+09:00Definitions and Propositions1898: Finite Composition of Quotient Maps Is Quotient, if Codomains of Constituent Maps Equal Domains of Succeeding Maps <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that finite composition of quotient maps is quotient, if codomains of constituent maps equal domains of succeeding maps Topics About: topological space The table of contents of this article Starting Context Target Context Orientation https://thebiasplanet.blogspot.com/2026/07/finitecompositionofquotientmapsisquotientifcodomainsofconstituentmapsequaldomainsofsucceedingmaps.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3915427547571136561Sun, 26 Jul 2026 12:51:56 +00002026-07-26T21:51:56.839+09:00Definitions and Propositions1897: For \(1\)-Dimensional Euclidean Topological Space and Equivalence Relation That \(2\) Elements Are Equivalent iff Their Difference Is Rational, Quotient Topology Is Trivial <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for \(1\)-dimensional Euclidean topological space and equivalence relation that \(2\) elements are equivalent iff their difference is rational, quotient topology is trivial Topics About: topological space The table of contents of https://thebiasplanet.blogspot.com/2026/07/for1dimensionaleuclideantopologicalspaceandequivalencerelationthat2elementsareequivalentifftheirdifferenceisrationalquotienttopologyistrivial.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7846163410332180539Sun, 26 Jul 2026 12:50:18 +00002026-07-26T21:50:18.300+09:00Definitions and Propositions1896: For Real Numbers Set and Equivalence Relation That \(2\) Elements Are Equivalent iff Their Difference Is Rational, Quotient Set Is Uncountable <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 numbers set and equivalence relation that \(2\) elements are equivalent iff their difference is rational, quotient set is uncountable Topics About: set The table of contents of this article Starting Context Target Context https://thebiasplanet.blogspot.com/2026/07/forrealnumberssetandequivalencerelationthat2elementsareequivalentifftheirdifferenceisrationalquotientsetisuncountable.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6417908381695817776Sun, 26 Jul 2026 12:48:45 +00002026-07-26T21:48:45.191+09:00Definitions and Propositions1895: Infinite Product of Sets Each of Which Has More than \(1\) Elements Is Uncountable <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that infinite product of sets each of which has more than \(1\) elements is uncountable Topics About: set The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note 3: https://thebiasplanet.blogspot.com/2026/07/infiniteproductofsetseachofwhichhasmorethan1elementsisuncountable.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6131131734575895830Sun, 26 Jul 2026 12:47:20 +00002026-07-26T21:47:20.061+09:00Definitions and Propositions1894: Finite Product of Countable Sets Is Countable <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of finite product of countable sets is countable 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 Context The reader knows https://thebiasplanet.blogspot.com/2026/07/finiteproductofcountablesetsiscountable.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3334725156154567167Sun, 26 Jul 2026 12:45:53 +00002026-07-26T21:45:53.487+09:00Definitions and Propositions1893: Finite Product Set Is 'Sets - Maps' Isomorphic to Sequential Products 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 finite product set is 'sets - maps' isomorphic to sequential products of sets 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/2026/07/finiteproductsetissetsmapsisomorphictosequentialproductsofsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4864953084935031255Sun, 26 Jul 2026 12:43:47 +00002026-07-26T21:43:47.335+09:00Definitions and Propositions1892: A Way to Systematically Choose Rational Number That Is Larger Than Real Number and Is Equal to or Smaller Than Another Real Number <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of a way to systematically choose rational number that is larger than real number and is equal to or smaller than another real number Topics About: set The table of contents of this article Starting Context Target Context Orientation Main https://thebiasplanet.blogspot.com/2026/07/awaytosystematicallychooserationalnumberthatislargerthanrealnumberandisequaltoorsmallerthananotherrealnumber.htmlnoreply@blogger.com (Unknown)