tag:blogger.com,1999:blog-922111353248289998Tue, 14 Apr 2026 22:31:28 +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)Blogger2220125tag:blogger.com,1999:blog-922111353248289998.post-7789324942804758534Sun, 12 Apr 2026 11:38:00 +00002026-04-12T20:38:22.710+09:00Definitions and Propositions1733: For Net with Directed Index Set Eventually in Subset of Topological Space That Converges on Space, Convergence Is on Closure of Subset <The previous article in this series | The table of contents of this series | description/proof of that for net with directed index set eventually in subset of topological space that converges on space, convergence is on closure of subset Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: https://thebiasplanet.blogspot.com/2026/04/fornetwithdirectedindexseteventuallyinsubsetoftopologicalspacethatconvergesonspaceconvergenceisonclosureofsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-487741868456903538Sun, 12 Apr 2026 11:36:00 +00002026-04-12T20:36:51.629+09:00Definitions and Propositions1732: Compact 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 compact 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 The https://thebiasplanet.blogspot.com/2026/04/compacthausdorfftopologicalspaceisnormal.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-746620447981080651Sun, 12 Apr 2026 11:35:00 +00002026-04-12T20:35:10.553+09:00Definitions and Propositions1731: Continuous Bijection from Compact Topological Space onto Hausdorff Topological Space Is Homeomorphism <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that continuous bijection from compact topological space onto Hausdorff topological space is homeomorphism Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: https://thebiasplanet.blogspot.com/2026/04/continuousbijectionfromcompacttopologicalspaceontohausdorfftopologicalspaceishomeomorphism.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2304997703557574593Sun, 12 Apr 2026 11:33:00 +00002026-04-12T20:33:36.605+09:00Definitions and Propositions1730: Closed Continuous Bijection Is Homeomorphism <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that closed continuous bijection is homeomorphism 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/2026/04/closedcontinuousbijectionishomeomorphism.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-238480632857434587Sun, 12 Apr 2026 11:31:00 +00002026-04-12T20:31:25.272+09:00Definitions and Propositions1729: Open Continuous Bijection Is Homeomorphism <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that open continuous bijection is homeomorphism 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/2026/04/opencontinuousbijectionishomeomorphism.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1265333646024573755Sun, 12 Apr 2026 11:29:00 +00002026-04-12T20:29:43.371+09:00Definitions and Propositions1728: For Topological Space, Subspace, and Point on Subspace, Intersection of Neighborhoods Basis at Point on Base Space and Subspace Is Neighborhoods Basis at Point on 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 topological space, subspace, and point on subspace, intersection of neighborhoods basis at point on base space and subspace is neighborhoods basis at point on subspace Topics About: topological space The table of contents of this https://thebiasplanet.blogspot.com/2026/04/fortopologicalspacesubspaceandpointonsubspaceintersectionofneighborhoodsbasisatpointonbasespaceandsubspaceisneighborhoodsbasisatpointonsubspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1478368667159553418Sun, 12 Apr 2026 11:28:00 +00002026-04-12T20:28:03.796+09:00Definitions and Propositions1727: Supremum of Lower Semicontinuous Maps Is Lower Semicontinuous <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that supremum of lower semicontinuous maps is lower semicontinuous 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/2026/04/supremumoflowersemicontinuousmapsislowersemicontinuous.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5295808724931253871Sun, 12 Apr 2026 11:26:00 +00002026-04-12T20:26:36.286+09:00Definitions and Propositions1726: Restriction of Lower Semicontinuous Map on Domain Is Lower Semicontinuous <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that restriction of lower semicontinuous map on domain is lower semicontinuous 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/2026/04/restrictionoflowersemicontinuousmapondomainislowersemicontinuous.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2086201540498878349Sun, 12 Apr 2026 11:25:00 +00002026-04-12T20:25:06.580+09:00Definitions and Propositions1725: For Lower Semicontinuous Map from Subspace of Product Topological Space into \(1\)-Dimensional Extended Euclidean Topological Space and Element of Subproduct, Cross Section of Map Is Lower Semicontinuous <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for lower semicontinuous map from subspace of product topological space into \(1\)-dimensional extended Euclidean topological space and element of subproduct, cross section of map is lower semicontinuous Topics About: topological space https://thebiasplanet.blogspot.com/2026/04/forlowersemicontinuousmapfromsubspaceofproducttopologicalspaceinto1dimensionalextendedeuclideantopologicalspaceandelementofsubproductcrosssectionofmapislowersemicontinuous.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2185846903733344069Sun, 12 Apr 2026 11:23:00 +00002026-04-12T20:23:36.248+09:00Definitions and Propositions1724: For Product Topological Space, Subspace, Open Subset of Subspace, and Element of Subproduct, Cross Section of Open Subset Is Open on Cross Section of 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 product topological space, subspace, open subset of subspace, and element of subproduct, cross section of open subset is open on cross section of subspace Topics About: topological space The table of contents of this article https://thebiasplanet.blogspot.com/2026/04/forproducttopologicalspacesubspaceopensubsetofsubspaceandelementofsubproductcrosssectionofopensubsetisopenoncrosssectionofsubspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7076742120904874153Sun, 12 Apr 2026 11:21:00 +00002026-04-12T20:21:18.963+09:00Definitions and Propositions1723: For Product Set, Subset, and Element of Subproduct, Cross Section of Subset Is Contained in Projection 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 product set, subset, and element of subproduct, cross section of subset is contained in projection of subset Topics About: set The table of contents of this article Starting Context Target Context Orientation Main Body 1: https://thebiasplanet.blogspot.com/2026/04/forproductsetsubsetandelementofsubproductcrosssectionofsubsetiscontainedinprojectionofsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6049273731232398531Sun, 12 Apr 2026 11:19:00 +00002026-04-12T20:19:51.944+09:00Definitions and Propositions1722: For Product Set, \(2\) Subsets, and Element of Subproduct, Cross Section of 1st Subset Minus 2nd Subset Is Cross Section of 1st Subset Minus Cross Section of 2nd 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 product set, \(2\) subsets, and element of subproduct, cross section of 1st subset minus 2nd subset is cross section of 1st subset minus cross section of 2nd subset Topics About: set The table of contents of this article Startinghttps://thebiasplanet.blogspot.com/2026/04/forproductset2subsetsandelementofsubproductcrosssectionof1stsubsetminus2ndsubsetiscrosssectionof1stsubsetminuscrosssectionof2ndsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1185375858830667084Sun, 05 Apr 2026 12:45:00 +00002026-04-12T20:18:07.268+09:00Definitions and Propositions1721: For Product Set, Subsets, and Element of Subproduct, Cross Section of Union of Subsets Is Union of Cross Sections 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 product set, subsets, and element of subproduct, cross section of union of subsets is union of cross sections of subsets Topics About: set The table of contents of this article Starting Context Target Context Orientation Main https://thebiasplanet.blogspot.com/2026/04/forproductsetsubsetsandelementofsubproductcrosssectionofunionofsubsetsisunionofcrosssectionsofsubsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8899889326815605778Sun, 05 Apr 2026 12:42:00 +00002026-04-05T21:42:50.948+09:00Definitions and Propositions1720: For Product Set, Subsets, and Element of Subproduct, Cross Section of Intersection of Subsets Is Intersection of Cross Sections 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 product set, subsets, and element of subproduct, cross section of intersection of subsets is intersection of cross sections of subsets Topics About: set The table of contents of this article Starting Context Target Context https://thebiasplanet.blogspot.com/2026/04/forproductsetsubsetsandelementofsubproductcrosssectionofintersectionofsubsetsisintersectionofcrosssectionsofsubsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2882376166001077672Sun, 05 Apr 2026 12:41:00 +00002026-04-05T21:41:26.357+09:00Definitions and Propositions1719: For Product Set, Subset, Subset That Contains 1st Subset, and Element of Subproduct, Cross Section of 1st Subset Is Contained in Cross Section of 2nd 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 product set, subset, subset that contains 1st subset, and element of subproduct, cross section of 1st subset is contained in cross section of 2nd subset Topics About: set The table of contents of this article Starting Context https://thebiasplanet.blogspot.com/2026/04/forproductsetsubsetsubsetthatcontains1stsubsetandelementofsubproductcrosssectionof1stsubsetiscontainedincrosssectionof2ndsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3499494596331657248Sun, 05 Apr 2026 12:39:00 +00002026-04-05T21:39:59.589+09:00Definitions and Propositions1718: Lower Semicontinuous Map from Topological Space into \(1\)-Dimensional Extended Euclidean Topological Space <The previous article in this series | The table of contents of this series | The next article in this series> definition of lower semicontinuous map from topological space into \(1\)-dimensional extended Euclidean topological space Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structuredhttps://thebiasplanet.blogspot.com/2026/04/lowersemicontinuousmapfromtopologicalspaceinto1dimensionalextendedeuclideantopologicalspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2949429741646533052Sun, 05 Apr 2026 12:38:00 +00002026-04-05T21:38:27.683+09:00Definitions and Propositions1717: Map from Topological Space into \(1\)-Dimensional Extended Euclidean Topological Space Lower Semicontinuous at Point <The previous article in this series | The table of contents of this series | The next article in this series> definition of map from topological space into \(1\)-dimensional extended Euclidean topological space lower semicontinuous at point Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: https://thebiasplanet.blogspot.com/2026/04/mapfromtopologicalspaceinto1dimensionalextendedeuclideantopologicalspacelowersemicontinuousatpoint.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4421373706052510769Sun, 05 Apr 2026 12:36:00 +00002026-04-05T21:36:55.947+09:00Definitions and Propositions1716: Projection of Product Topological Space Is Not Necessarily Closed <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that projection of product topological space is not necessarily closed 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 https://thebiasplanet.blogspot.com/2026/04/projectionofproducttopologicalspaceisnotnecessarilyclosed.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2634403255072881115Sun, 05 Apr 2026 12:35:00 +00002026-04-05T21:35:29.768+09:00Definitions and Propositions1715: Projection of Product Topological Space Is Open <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that projection of product topological space is open 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 Contexthttps://thebiasplanet.blogspot.com/2026/04/projectionofproducttopologicalspaceisopen.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3081752963439742185Sun, 05 Apr 2026 12:34:00 +00002026-04-05T21:34:00.561+09:00Definitions and Propositions1714: For Product Set and Subsets, Projection of Union of Subsets Is Union of Projections 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 product set and subsets, projection of union of subsets is union of projections of subsets Topics About: set The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2:https://thebiasplanet.blogspot.com/2026/04/forproductsetandsubsetsprojectionofunionofsubsetsisunionofprojectionsofsubsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3381335771785830808Sun, 05 Apr 2026 12:32:00 +00002026-04-05T21:32:32.588+09:00Definitions and Propositions1713: For Product Set and Subsets, Projection of Intersection of Subsets Is Contained in But Not Necessarily Is Intersection of Projections 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 product set and subsets, projection of intersection of subsets is contained in but not necessarily is intersection of projections of subsets Topics About: set The table of contents of this article Starting Context Target Context https://thebiasplanet.blogspot.com/2026/04/forproductsetandsubsetsprojectionofintersectionofsubsetsiscontainedinbutnotnecessarilyisintersectionofprojectionsofsubsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4971115816467576614Sun, 05 Apr 2026 12:30:00 +00002026-04-05T21:30:55.670+09:00Definitions and Propositions1712: Cross Section of Map from Subset of Product Set by Element of Subproduct Set <The previous article in this series | The table of contents of this series | The next article in this series> definition of cross section of map from subset of product set by element of subproduct 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 https://thebiasplanet.blogspot.com/2026/04/crosssectionofmapfromsubsetofproductsetbyelementofsubproductset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3178043964428138374Sun, 05 Apr 2026 12:29:00 +00002026-04-05T21:29:25.641+09:00Definitions and Propositions1711: Cross Section of Subset of Product Set by Element of Subproduct Set <The previous article in this series | The table of contents of this series | The next article in this series> definition of cross section of subset of product set by element of subproduct 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 https://thebiasplanet.blogspot.com/2026/04/crosssectionofsubsetofproductsetbyelementofsubproductset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-2230255750756440742Sun, 05 Apr 2026 12:27:00 +00002026-04-05T21:27:59.860+09:00Definitions and Propositions1710: Projection from Product Set onto Subproduct Set <The previous article in this series | The table of contents of this series | The next article in this series> definition of projection from product set onto subproduct 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 https://thebiasplanet.blogspot.com/2026/04/projectionfromproductsetontosubproductset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3191692267805305469Sun, 05 Apr 2026 12:26:00 +00002026-04-05T21:26:27.002+09:00Definitions and Propositions1709: Sum of Uncountable Number of Positive Real Numbers Is Infinity <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that sum of uncountable number of positive real numbers is infinity Topics About: metric 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/2026/04/sumofuncountablenumberofpositiverealnumbersisinfinity.htmlnoreply@blogger.com (Unknown)