tag:blogger.com,1999:blog-922111353248289998Sun, 12 Jul 2026 12:18:44 +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)Blogger2366125tag:blogger.com,1999:blog-922111353248289998.post-2854970580294147596Sun, 12 Jul 2026 12:18:44 +00002026-07-12T21:18:44.398+09:00Definitions and Propositions1879: For Containing Convergent Sequence of Subsets of Real Numbers Set with Canonical Ordering, Supremum of Containing Convergence Is Convergence of Sequence of Supremums of Subsets <The previous article in this series | The table of contents of this series | description/proof of that for containing convergent sequence of subsets of real numbers set with canonical ordering, supremum of containing convergence is convergence of sequence of supremums of subsets Topics About: metric space The table of contents of this article Starting Context Target https://thebiasplanet.blogspot.com/2026/07/forcontainingconvergentsequenceofsubsetsofrealnumberssetwithcanonicalorderingsupremumofcontainingconvergenceisconvergenceofsequenceofsupremumsofsubsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8654118633913181752Sun, 12 Jul 2026 12:17:09 +00002026-07-12T21:17:09.123+09:00Definitions and Propositions1878: For Containing Convergent Sequence of Subsets of Real Numbers Set with Canonical Ordering, Infimum of Containing Convergence Is Convergence of Sequence of Infimums 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 containing convergent sequence of subsets of real numbers set with canonical ordering, infimum of containing convergence is convergence of sequence of infimums of subsets Topics About: metric space The table of contents of this https://thebiasplanet.blogspot.com/2026/07/forcontainingconvergentsequenceofsubsetsofrealnumberssetwithcanonicalorderinginfimumofcontainingconvergenceisconvergenceofsequenceofinfimumsofsubsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4148489342518389401Sun, 12 Jul 2026 12:15:35 +00002026-07-12T21:15:35.205+09:00Definitions and Propositions1877: Contained Convergence of Map from Open Interval into Power Set of Set at Boundary <The previous article in this series | The table of contents of this series | The next article in this series> definition of contained convergence of map from open interval into power set of set at boundary Topics About: set 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/2026/07/containedconvergenceofmapfromopenintervalintopowersetofsetatboundary.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3465020867097034425Sun, 12 Jul 2026 12:14:08 +00002026-07-12T21:14:08.229+09:00Definitions and Propositions1876: Containing Convergence of Map from Open Interval into Power Set of Set at Boundary <The previous article in this series | The table of contents of this series | The next article in this series> definition of containing convergence of map from open interval into power set of set at boundary Topics About: set The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note Starting Contexthttps://thebiasplanet.blogspot.com/2026/07/containingconvergenceofmapfromopenintervalintopowersetofsetatboundary.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2952938553251120394Sun, 12 Jul 2026 12:12:35 +00002026-07-12T21:12:35.445+09:00Definitions and Propositions1875: Contained Convergence of Sequence of Subsets of Set <The previous article in this series | The table of contents of this series | The next article in this series> definition of contained convergence of sequence of subsets of 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 https://thebiasplanet.blogspot.com/2026/07/containedconvergenceofsequenceofsubsetsofset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5696498781884807552Sun, 12 Jul 2026 12:11:08 +00002026-07-12T21:11:08.995+09:00Definitions and Propositions1874: Containing Convergence of Sequence of Subsets of Set <The previous article in this series | The table of contents of this series | The next article in this series> definition of containing convergence of sequence of subsets of 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 https://thebiasplanet.blogspot.com/2026/07/containingconvergenceofsequenceofsubsetsofset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3338664250952982391Sun, 12 Jul 2026 12:09:39 +00002026-07-12T21:09:39.876+09:00Definitions and Propositions1873: For Non-Negative Measurable Extended Real Function over Finite Measure Space, Function Is Integrable iff Sum of Measures of Preimages of Lower-Closed-Positive-Natural-Number-Bounded Intervals Converges <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 extended real function over finite measure space, function is integrable iff sum of measures of preimages of lower-closed-positive-natural-number-bounded intervals converges Topics About: measure space The https://thebiasplanet.blogspot.com/2026/07/fornonnegativemeasurableextendedrealfunctionoverfinitemeasurespacefunctionisintegrableiffsumofmeasuresofpreimagesoflowerclosedpositivenaturalnumberboundedintervalsconverges.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-698289635390396777Sun, 12 Jul 2026 12:08:16 +00002026-07-12T21:08:16.770+09:00Definitions and Propositions1872: For Measurable Extended Real Function over Measure Space Whose Range Is in Union of Natural Numbers Set and Infinity, Lebesgue Integral of Map Is Sum of Measures of Preimages of Closed-Positive-Natural-Number-or-Infinity-Lower-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 for measurable extended real function over measure space whose range is in union of natural numbers set and infinity, Lebesgue integral of map is sum of measures of preimages of closed-positive-natural-number-or-infinity-lower-bounded intervals https://thebiasplanet.blogspot.com/2026/07/formeasurableextendedrealfunctionovermeasurespacewhoserangeisinunionofnaturalnumberssetandinfinitylebesgueintegralofmapissumofmeasuresofpreimagesofclosedpositivenaturalnumberorinfinitylowerboundedintervals.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8222826148202024409Sun, 12 Jul 2026 12:06:41 +00002026-07-12T21:06:41.371+09:00Definitions and Propositions1871: For Measure Space and Measurable Subset, Lebesgue Integral of Integrable Complex Function over Space Is Integral over Subset Plus Integral over 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 measure space and measurable subset, Lebesgue integral of integrable complex function over space is integral over subset plus integral over complement of subset Topics About: measure space The table of contents of this article https://thebiasplanet.blogspot.com/2026/07/formeasurespaceandmeasurablesubsetlebesgueintegralofintegrablecomplexfunctionoverspaceisintegraloversubsetplusintegralovercomplementofsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3424777569608270925Sun, 05 Jul 2026 12:40:54 +00002026-07-12T21:05:00.212+09:00Definitions and Propositions1870: For Measure Space and Measurable Subset, Lebesgue Integral of Measurable Extended Real Function over Space Is Integral over Subset Plus Integral over 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 measure space and measurable subset, Lebesgue integral of measurable extended real function over space is integral over subset plus integral over complement of subset Topics About: measure space The table of contents of this https://thebiasplanet.blogspot.com/2026/07/formeasurespaceandmeasurablesubsetlebesgueintegralofmeasurableextendedrealfunctionoverspaceisintegraloversubsetplusintegralovercomplementofsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6123426592888835205Sun, 05 Jul 2026 12:39:31 +00002026-07-05T21:39:31.956+09:00Definitions and Propositions1869: Topological Space Is Compact iff for Each Set of Closed Subsets Whose Intersection Is Empty, There Is Nonempty Finite Subset Whose Intersection Is Empty <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 compact iff for each set of closed subsets whose intersection is empty, there is nonempty finite subset whose intersection is empty Topics About: topological space The table of contents of this article Starting https://thebiasplanet.blogspot.com/2026/07/topologicalspaceiscompactiffforeachsetofclosedsubsetswhoseintersectionisemptythereisnonemptyfinitesubsetwhoseintersectionisempty.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8791676175875764472Sun, 05 Jul 2026 12:38:11 +00002026-07-05T21:38:11.598+09:00Definitions and Propositions1868: For Map from Metric Space Minus Point into Metric Space, iff for Each Sequence on Domain That Converges to Point, Its Image Converges to Codomain Point, Map Converges w.r.t. Point to Codomain Point <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 metric space minus point into metric space, iff for each sequence on domain that converges to point, its image converges to codomain point, map converges w.r.t. point to codomain point Topics About: metric space The tablehttps://thebiasplanet.blogspot.com/2026/07/formapfrommetricspaceminuspointintometricspaceiffforeachsequenceondomainthatconvergestopointitsimageconvergestocodomainpointmapconvergeswrtpointtocodomainpoint.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2920399557061415919Sun, 05 Jul 2026 12:36:44 +00002026-07-05T21:36:44.786+09:00Definitions and Propositions1867: For Map Between Metric Spaces and Domain Point, iff for Each Sequence on Domain That Converges to Point, Its Image Converges to Image of Point, Map Is Continuous at Point <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 between metric spaces and domain point, iff for each sequence on domain that converges to point, its image converges to image of point, map is continuous at point Topics About: metric space The table of contents of this https://thebiasplanet.blogspot.com/2026/07/formapbetweenmetricspacesanddomainpointiffforeachsequenceondomainthatconvergestopointitsimageconvergestoimageofpointmapiscontinuousatpoint.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7060338765779338538Sun, 05 Jul 2026 12:35:19 +00002026-07-05T21:35:19.111+09:00Definitions and Propositions1866: For \(1\)-Dimensional Euclidean Topological Space and Open Subset, Open Subset Is Disjoint Union of Countable 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 for \(1\)-dimensional Euclidean topological space and open subset, open subset is disjoint union of countable open intervals Topics About: topological space The table of contents of this article Starting Context Target Context https://thebiasplanet.blogspot.com/2026/07/for1dimensionaleuclideantopologicalspaceandopensubsetopensubsetisdisjointunionofcountableopenintervals.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6759438836138431813Sun, 05 Jul 2026 12:33:52 +00002026-07-05T21:33:52.645+09:00Definitions and Propositions1865: For Infinite Set, if There Is Injection from Set into Natural Numbers Set, There Is Bijection from Natural Numbers Set onto 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 infinite set, if there is injection from set into natural numbers set, there is bijection from natural numbers set onto set Topics About: set The table of contents of this article Starting Context Target Context Orientation Main https://thebiasplanet.blogspot.com/2026/07/forinfinitesetifthereisinjectionfromsetintonaturalnumberssetthereisbijectionfromnaturalnumberssetontoset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-373403033496851779Sun, 05 Jul 2026 12:32:19 +00002026-07-05T22:31:07.966+09:00Definitions and Propositions1864: For Continuous Map Between Topological Spaces, Boundary of Preimage of Subset Is Contained in but Not Necessarily Equal to Preimage of Boundary 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 continuous map between topological spaces, boundary of preimage of subset is contained in but not necessarily equal to preimage of boundary of subset Topics About: topological space The table of contents of this article Starting https://thebiasplanet.blogspot.com/2026/07/forcontinuousmapbetweentopologicalspacesboundaryofpreimageofsubsetiscontainedinbutnotnecessarilyequaltopreimageofboundaryofsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7983900120418518313Sun, 05 Jul 2026 12:30:49 +00002026-07-05T21:30:49.002+09:00Definitions and Propositions1863: For Continuous Map Between Topological Spaces, Preimage of Interior of Subset Is Contained in but Not Necessarily Equal to Interior of Preimage 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 continuous map between topological spaces, preimage of interior of subset is contained in but not necessarily equal to interior of preimage of subset Topics About: topological space The table of contents of this article Starting https://thebiasplanet.blogspot.com/2026/07/forcontinuousmapbetweentopologicalspacespreimageofinteriorofsubsetiscontainedinbutnotnecessarilyequaltointeriorofpreimageofsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1004181027540828274Sun, 05 Jul 2026 12:29:19 +00002026-07-05T21:29:19.482+09:00Definitions and Propositions1862: For Sequence on \(1\)-Dimensional Euclidean Metric Space with Canonical Ordering, if Limit Inferior and Limit Superior Exist and Are Equal, Convergence Is Limit Inferior and Limit Superior <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for sequence on \(1\)-dimensional Euclidean metric space with canonical ordering, if limit inferior and limit superior exist and are equal, convergence is limit inferior and limit superior Topics About: metric space The table of https://thebiasplanet.blogspot.com/2026/07/forsequenceon1dimensionaleuclideanmetricspacewithcanonicalorderingiflimitinferiorandlimitsuperiorexistandareequalconvergenceislimitinferiorandlimitsuperior.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4641351859049045502Sun, 05 Jul 2026 12:27:46 +00002026-07-05T22:30:24.053+09:00Definitions and Propositions1861: For Sequence on \(1\)-Dimensional Euclidean Metric Space with Canonical Ordering, if Convergence Exists, Convergence Is Limit Inferior and Limit Superior <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for sequence on \(1\)-dimensional Euclidean metric space with canonical ordering, if convergence exists, convergence is limit inferior and limit superior Topics About: metric space The table of contents of this article Starting https://thebiasplanet.blogspot.com/2026/07/forsequenceon1dimensionaleuclideanmetricspacewithcanonicalorderingifconvergenceexistsconvergenceislimitinferiorandlimitsuperior.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6475683218814592416Sun, 05 Jul 2026 12:26:06 +00002026-07-05T21:26:06.498+09:00Definitions and Propositions1860: For Sequence on Real Numbers Set with Canonical Ordering, if Limit Inferior and Limit Superior Exist, Limit Inferior Is Equal to or Smaller than Limit Superior <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for sequence on real numbers set with canonical ordering, if limit inferior and limit superior exist, limit inferior is equal to or smaller than limit superior Topics About: set The table of contents of this article Starting Context https://thebiasplanet.blogspot.com/2026/07/forsequenceonrealnumberssetwithcanonicalorderingiflimitinferiorandlimitsuperiorexistlimitinferiorisequaltoorsmallerthanlimitsuperior.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4254296573885659721Sun, 05 Jul 2026 12:24:25 +00002026-07-05T21:24:25.047+09:00Definitions and Propositions1859: For Value-Bounded Sequence on Real Numbers Set, Limit Inferior and Limit Superior Exist <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for value-bounded sequence on real numbers set, limit inferior and limit superior exist Topics About: set The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Note 3https://thebiasplanet.blogspot.com/2026/07/forvalueboundedsequenceonrealnumberssetlimitinferiorandlimitsuperiorexist.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-623192658516467845Sun, 05 Jul 2026 12:22:48 +00002026-07-05T21:22:48.340+09:00Definitions and Propositions1858: For Sequence on Real Numbers Set with Canonical Ordering, if Limit Superior Exists, Limit Inferior Does Not Necessarily Exist, and if Limit Inferior Exists, Limit Superior Does Not Necessarily Exist <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for sequence on real numbers set with canonical ordering, if limit superior exists, limit inferior does not necessarily exist, and if limit inferior exists, limit superior does not necessarily exist Topics About: set The table of https://thebiasplanet.blogspot.com/2026/07/forsequenceonrealnumberssetwithcanonicalorderingiflimitsuperiorexistslimitinferiordoesnotnecessarilyexistandiflimitinferiorexistslimitsuperiordoesnotnecessarilyexist.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4235336852968332121Sun, 28 Jun 2026 13:34:06 +00002026-07-05T21:21:09.777+09:00Definitions and Propositions1857: For Partially-Ordered Set and Nonempty Subset, if Infimum and Supremum of Subset Exist, Infimum Is Equal to or Smaller than Supremum <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for partially-ordered set and nonempty subset, if infimum and supremum of subset exist, infimum is equal to or smaller than supremum Topics About: set The table of contents of this article Starting Context Target Context Orientation https://thebiasplanet.blogspot.com/2026/06/forpartiallyorderedsetandnonemptysubsetifinfimumandsupremumofsubsetexistinfimumisequaltoorsmallerthansupremum.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5473293552496899694Sun, 28 Jun 2026 13:32:25 +00002026-06-28T22:32:25.818+09:00Definitions and Propositions1856: For Non-Decreasing and Non-Increasing Sequences on Real Numbers Set s.t. 1st Sequence Is Equal to or Smaller than 2nd Sequence, Each Element of 1st Sequence Is Equal to or Smaller than Any Element of 2nd Sequence, and Supremum of 1st Sequence Is Equal to or Smaller than Infimum of 2nd 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 and non-increasing sequences on real numbers set s.t. 1st sequence is equal to or smaller than 2nd sequence, each element of 1st sequence is equal to or smaller than any element of 2nd sequence, and supremum of 1st sequence is equalhttps://thebiasplanet.blogspot.com/2026/06/fornondecreasingandnonincreasingsequencesonrealnumberssetst1stsequenceisequaltoorsmallerthan2ndsequenceeachelementof1stsequenceisequaltoorsmallerthananyelementof2ndsequenceandsupremumof1stsequenceisequaltoorsmallerthaninfimumof2ndsequence.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8872870704160526496Sun, 28 Jun 2026 13:30:50 +00002026-06-28T22:30:50.003+09:00Definitions and Propositions1855: For \(2\) Sequences on Partially-Ordered Ring with Same Domain, Limit Superior of Sum of Sequences Is Not Necessarily Sum of Limits Superior of Sequences, and Limit Inferior of Sum of Sequences Is Not Necessarily Sum of Limits Inferior of Sequences <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for \(2\) sequences on partially-ordered ring with same domain, limit superior of sum of sequences is not necessarily sum of limits superior of sequences, and limit inferior of sum of sequences is not necessarily sum of limits inferior of sequences https://thebiasplanet.blogspot.com/2026/06/for2sequencesonpartiallyorderedringwithsamedomainlimitsuperiorofsumofsequencesisnotnecessarilysumoflimitssuperiorofsequencesandlimitinferiorofsumofsequencesisnotnecessarilysumoflimitsinferiorofsequences.htmlnoreply@blogger.com (Unknown)