tag:blogger.com,1999:blog-922111353248289998Sun, 21 Jun 2026 12:46:20 +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)Blogger2331125tag:blogger.com,1999:blog-922111353248289998.post-8740963538772092300Sun, 21 Jun 2026 12:46:20 +00002026-06-21T21:46:20.881+09:00Definitions and Propositions1844: 2nd-Countable Locally Compact Hausdorff Topological Space Is \(\sigma\)-Compact and Paracompact <The previous article in this series | The table of contents of this series | description/proof of that 2nd-countable locally compact Hausdorff topological space is \(\sigma\)-compact and paracompact Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Proof Starting https://thebiasplanet.blogspot.com/2026/06/2ndcountablelocallycompacthausdorfftopologicalspaceissigmacompactandparacompact.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6215648457205906334Sun, 21 Jun 2026 12:44:32 +00002026-06-21T21:44:32.958+09:00Definitions and Propositions1843: For Paracompact Hausdorff Topological Space, Open Cover of Space, and Locally Finite Refinement, There Is Partition of Unity Subordinate to Refinement <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of for paracompact Hausdorff topological space, open cover of space, and locally finite refinement, there is partition of unity subordinate to refinement Topics About: topological space The table of contents of this article Starting Context https://thebiasplanet.blogspot.com/2026/06/forparacompacthausdorfftopologicalspaceopencoverofspaceandlocallyfiniterefinementthereispartitionofunitysubordinatetorefinement.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5614990473687911656Sun, 21 Jun 2026 12:43:07 +00002026-06-21T21:43:07.083+09:00Definitions and Propositions1842: For Linearly-Ordered Set, Finite Subset Can Be Ordered in Line <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of for linearly-ordered set, finite subset can be ordered in line Topics About: set 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/2026/06/forlinearlyorderedsetfinitesubsetcanbeorderedinline.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2485208349228356360Sun, 21 Jun 2026 12:41:47 +00002026-06-21T21:41:47.081+09:00Definitions and Propositions1841: For Topological Space and Set of Continuous Maps from Space into Euclidean Topological Space s.t. Set of Preimages of Nonzero Is Locally Finite, Sum of Maps Is Continuous <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of for topological space and set of continuous maps from space into Euclidean topological space s.t. set of preimages of nonzero is locally finite, sum of maps is continuous Topics About: topological space The table of contents of this https://thebiasplanet.blogspot.com/2026/06/fortopologicalspaceandsetofcontinuousmapsfromspaceintoeuclideantopologicalspacestsetofpreimagesofnonzeroislocallyfinitesumofmapsiscontinuous.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7907377273419802523Sun, 21 Jun 2026 12:40:17 +00002026-06-21T21:40:17.789+09:00Definitions and Propositions1840: For Set and Subset, for Finite Cover of Subset, There Is Subcover Whose Each Element Is Indispensable, and for Infinite Cover of Subset, There Is Not Necessarily Subcover Whose Each Element Is Indispensable <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 subset, for finite cover of subset, there is subcover whose each element is indispensable, and for infinite cover of subset, there is not necessarily subcover whose each element is indispensable Topics About: set The tablehttps://thebiasplanet.blogspot.com/2026/06/forsetandsubsetforfinitecoverofsubsetthereissubcoverwhoseeachelementisindispensableandforinfinitecoverofsubsetthereisnotnecessarilysubcoverwhoseeachelementisindispensable.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7840849411799741047Sun, 21 Jun 2026 12:38:58 +00002026-06-21T21:38:58.428+09:00Definitions and Propositions1839: Set with Ordering as Containment Is Partially-Ordered Set <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that set with ordering as containment is partially-ordered set Topics About: set 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/2026/06/setwithorderingascontainmentispartiallyorderedset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-255730361030629354Sun, 21 Jun 2026 12:37:30 +00002026-06-21T21:37:30.541+09:00Definitions and Propositions1838: For Countable Set, Set of Finite Subsets Is Countable <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of for countable set, set of finite subsets is countable Topics About: set The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Proof Starting Context The reader knows https://thebiasplanet.blogspot.com/2026/06/forcountablesetsetoffinitesubsetsiscountable.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-557118342709804875Sun, 21 Jun 2026 12:36:05 +00002026-06-21T21:36:05.120+09:00Definitions and Propositions1837: Subset of Countable Set Is Countable <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of subset of countable set is countable Topics About: set The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Proof Starting Context The reader knows a definition of https://thebiasplanet.blogspot.com/2026/06/subsetofcountablesetiscountable.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2525871900472432382Sun, 21 Jun 2026 12:34:30 +00002026-06-21T21:34:30.477+09:00Definitions and Propositions1836: For Infinite Set, if There Is Surjection from Natural Numbers Set onto 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 surjection from natural numbers set onto 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 Mainhttps://thebiasplanet.blogspot.com/2026/06/forinfinitesetifthereissurjectionfromnaturalnumberssetontosetthereisbijectionfromnaturalnumberssetontoset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6924062754345573033Sun, 21 Jun 2026 12:33:09 +00002026-06-21T21:33:09.501+09:00Definitions and Propositions1835: Countable Set <The previous article in this series | The table of contents of this series | The next article in this series> definition of countable 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 bijection. Target Context https://thebiasplanet.blogspot.com/2026/06/countableset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3997887538266901208Sun, 14 Jun 2026 14:30:28 +00002026-06-21T21:31:46.516+09:00Definitions and Propositions1834: For Topological Space and Open Cover of Space, if There Are Locally Finite Refinement of Open Cover and Partition of Unity Subordinate to Refinement, There Is Partition of Unity Subordinate to Original Cover <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 and open cover of space, if there are locally finite refinement of open cover and partition of unity subordinate to refinement, there is partition of unity subordinate to original cover Topics About: topological space https://thebiasplanet.blogspot.com/2026/06/fortopologicalspaceandopencoverofspaceiftherearelocallyfiniterefinementofopencoverandpartitionofunitysubordinatetorefinementthereispartitionofunitysubordinatetooriginalcover.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7194390507805805916Sun, 14 Jun 2026 14:28:48 +00002026-06-14T23:28:48.481+09:00Definitions and Propositions1833: For Topological Space and Set of Maps from Space into Ring or Module s.t. Set of Preimages of Nonzero Is Locally Finite, Support of Sum of Maps Is Contained in Union of Supports of Maps <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 and set of maps from space into ring or module s.t. set of preimages of nonzero is locally finite, support of sum of maps is contained in union of supports of maps Topics About: topological space The table of https://thebiasplanet.blogspot.com/2026/06/fortopologicalspaceandsetofmapsfromspaceintoringormodulestsetofpreimagesofnonzeroislocallyfinitesupportofsumofmapsiscontainedinunionofsupportsofmaps.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6636504307977625108Sun, 14 Jun 2026 14:27:22 +00002026-06-14T23:27:22.118+09:00Definitions and Propositions1832: Paracompact Hausdorff Topological Space Is Normal <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that paracompact Hausdorff topological space is normal 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/2026/06/paracompacthausdorfftopologicalspaceisnormal.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8780549482070793498Sun, 14 Jun 2026 14:25:57 +00002026-06-14T23:25:57.045+09:00Definitions and Propositions1831: Closed Subspace of Paracompact Topological Space Is Paracompact <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that closed subspace of paracompact topological space is paracompact Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Proof 3: Note https://thebiasplanet.blogspot.com/2026/06/closedsubspaceofparacompacttopologicalspaceisparacompact.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4566999813960544960Sun, 14 Jun 2026 14:24:34 +00002026-06-14T23:24:34.801+09:00Definitions and Propositions1830: Paracompact Topological Space <The previous article in this series | The table of contents of this series | The next article in this series> definition of paracompact topological space Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description Starting Context The reader knows a definition of locally https://thebiasplanet.blogspot.com/2026/06/paracompacttopologicalspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5201865388898640866Sun, 14 Jun 2026 14:23:14 +00002026-06-14T23:23:14.925+09:00Definitions and Propositions1829: Refinement of Open Cover of Subset of Topological Space <The previous article in this series | The table of contents of this series | The next article in this series> definition of refinement of open cover of subset of 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 https://thebiasplanet.blogspot.com/2026/06/refinementofopencoverofsubsetoftopologicalspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5419958439674236654Sun, 14 Jun 2026 14:21:45 +00002026-06-14T23:21:45.441+09:00Definitions and Propositions1828: Locally Compact Hausdorff Topological Space Is Open Subspace of Its \(1\)-Point Compactification <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that locally compact Hausdorff topological space is open subspace of its \(1\)-point compactification Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: https://thebiasplanet.blogspot.com/2026/06/locallycompacthausdorfftopologicalspaceisopensubspaceofits1pointcompactification.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4282026382131223280Sun, 14 Jun 2026 14:20:16 +00002026-06-14T23:20:16.038+09:00Definitions and Propositions1827: For Hausdorff Topological Space, if There Is Locally Compact Hausdorff Topological Space of Which Space Is Locally Closed Subspace, Space Is Locally Compact <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for Hausdorff topological space, if there is locally compact Hausdorff topological space of which space is locally closed subspace, space is locally compact Topics About: topological space The table of contents of this article https://thebiasplanet.blogspot.com/2026/06/forhausdorfftopologicalspaceifthereislocallycompacthausdorfftopologicalspaceofwhichspaceislocallyclosedsubspacespaceislocallycompact.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1964808494248119348Sun, 14 Jun 2026 14:18:38 +00002026-06-14T23:18:38.266+09:00Definitions and Propositions1826: Locally Closed Topological Subspace <The previous article in this series | The table of contents of this series | The next article in this series> definition of locally closed topological subspace 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 definitionhttps://thebiasplanet.blogspot.com/2026/06/locallyclosedtopologicalsubspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6239231659861042027Sun, 14 Jun 2026 14:16:52 +00002026-06-14T23:16:52.034+09:00Definitions and Propositions1825: Proper Continuous Map Between Locally Compact Hausdorff Topological Spaces Is Closed <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that proper continuous map between locally compact Hausdorff topological spaces is closed Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured https://thebiasplanet.blogspot.com/2026/06/propercontinuousmapbetweenlocallycompacthausdorfftopologicalspacesisclosed.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4366843260408430753Sun, 14 Jun 2026 14:15:06 +00002026-06-14T23:15:06.771+09:00Definitions and Propositions1824: For Map and Its Extension That Maps Extended Area Outside Original Codomain, Original Map Image of Subset of Original Domain Is Intersection of Extension Image of Union of Subset and Extended Area and Original Codomain <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 its extension that maps extended area outside original codomain, original map image of subset of original domain is intersection of extension image of union of subset and extended area and original codomain Topics About: set https://thebiasplanet.blogspot.com/2026/06/formapanditsextensionthatmapsextendedareaoutsideoriginalcodomainoriginalmapimageofsubsetoforiginaldomainisintersectionofextensionimageofunionofsubsetandextendedareaandoriginalcodomain.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3563972229265081969Sun, 14 Jun 2026 14:12:48 +00002026-06-14T23:12:48.095+09:00Definitions and Propositions1823: For Map and Its Extension That Maps Extended Area Outside Original Codomain, Extension Preimage of Subset Is Union of Original Map Preimage of Intersection of Subset and Original Codomain and Extension Preimage of Subset Minus Original Codomain <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 its extension that maps extended area outside original codomain, extension preimage of subset is union of original map preimage of intersection of subset and original codomain and extension preimage of subset minus original codomain https://thebiasplanet.blogspot.com/2026/06/formapanditsextensionthatmapsextendedareaoutsideoriginalcodomainextensionpreimageofsubsetisunionoforiginalmappreimageofintersectionofsubsetandoriginalcodomainandextensionpreimageofsubsetminusoriginalcodomain.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5078624250205582550Sun, 14 Jun 2026 14:11:19 +00002026-06-14T23:11:19.587+09:00Definitions and Propositions1822: Map Preimage of Empty Subset Is Empty Set <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that map preimage of empty subset is empty set Topics About: set The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Proof Starting Context The reader knows a https://thebiasplanet.blogspot.com/2026/06/mappreimageofemptysubsetisemptyset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4760914176412697453Sun, 14 Jun 2026 14:09:57 +00002026-06-14T23:09:57.854+09:00Definitions and Propositions1821: Map Preimage of Subset of Codomain <The previous article in this series | The table of contents of this series | The next article in this series> definition of map preimage of subset of codomain Topics About: set The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description Starting Context The reader knows a definition of map. Target https://thebiasplanet.blogspot.com/2026/06/mappreimageofsubsetofcodomain.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3295400733108026902Sun, 14 Jun 2026 14:08:23 +00002026-06-14T23:08:23.238+09:00Definitions and Propositions1820: For Locally Compact Hausdorff Topological Space, Topology of \(1\)-Point Compactification Is Only Topology That Makes \(1\)-Point-Augmented Set Compact Hausdorff with Original Space as Subspace <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for locally compact Hausdorff topological space, topology of \(1\)-point compactification is only topology that makes \(1\)-point-augmented set compact Hausdorff with original space as subspace Topics About: topological space The https://thebiasplanet.blogspot.com/2026/06/forlocallycompacthausdorfftopologicalspacetopologyof1pointcompactificationisonlytopologythatmakes1pointaugmentedsetcompacthausdorffwithoriginalspaceassubspace.htmlnoreply@blogger.com (Unknown)