tag:blogger.com,1999:blog-922111353248289998Tue, 05 May 2026 22:38:34 +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)Blogger2254125tag:blogger.com,1999:blog-922111353248289998.post-5895490153985891445Mon, 04 May 2026 11:45:00 +00002026-05-04T20:45:13.122+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 | 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 Starting Context The reader knows a https://thebiasplanet.blogspot.com/2026/05/finiteproductoftopologicalsumsistopologicalsumofproducts.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1488594167589038545Mon, 04 May 2026 11:43:00 +00002026-05-04T20:43:34.026+09:00Definitions and Propositions1766: Injective \(C^\infty\) Immersion Is \(C^\infty\) Embedding iff Restriction of Map on Range Codomain 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 injective \(C^\infty\) immersion is C^\infty embedding iff restriction of map on range codomain is open Topics About: \(C^\infty\) manifold The table of contents of this article Starting Context Target Context Orientation Main Body 1https://thebiasplanet.blogspot.com/2026/05/injectivecinftyimmersioniscinftyembeddingiffrestrictionofmaponrangecodomainisopen.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8547269746029552004Mon, 04 May 2026 11:42:00 +00002026-05-04T20:42:00.715+09:00Definitions and Propositions1765: For Metric Space with Induced Topology, Compact Subset, and Open Cover of Subset, There Is Positive Real Number (Lebesgue Number) s.t. Subset of Subset with Diameter Smaller than Number Is Contained in Element of 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 metric space with induced topology, compact subset, and open cover of subset, there is positive real number (Lebesgue number) s.t. subset of subset with diameter smaller than number is contained in element of cover Topics About: metric https://thebiasplanet.blogspot.com/2026/05/formetricspacewithinducedtopologycompactsubsetandopencoverofsubsetthereispositiverealnumberlebesguenumberstsubsetofsubsetwithdiametersmallerthannumberiscontainedinelementofcover.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5725274217268772377Mon, 04 May 2026 11:40:00 +00002026-05-04T20:40:22.575+09:00Definitions and Propositions1764: Subspace of Completely Regular 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 subspace of completely regular 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 https://thebiasplanet.blogspot.com/2026/05/subspaceofcompletelyregulartopologicalspaceiscompletelyregular.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-9174541189052885228Mon, 04 May 2026 11:38:00 +00002026-05-04T20:38:25.310+09:00Definitions and Propositions1763: For Finite-Product Topological Space, Product of Constituent Subbases Is Subbasis <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for finite-product topological space, product of constituent subbases is subbasis Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2https://thebiasplanet.blogspot.com/2026/05/forfiniteproducttopologicalspaceproductofconstituentsubbasesissubbasis.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5738043557909152204Mon, 04 May 2026 11:36:00 +00002026-05-04T20:36:52.005+09:00Definitions and Propositions1762: For Finite-Product Topological Space, Product of Constituent Bases Is Basis <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for finite-product topological space, product of constituent bases is basis Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured Description 2: Notehttps://thebiasplanet.blogspot.com/2026/05/forfiniteproducttopologicalspaceproductofconstituentbasesisbasis.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-864746551635916293Mon, 04 May 2026 11:35:00 +00002026-05-04T20:35:15.950+09:00Definitions and Propositions1761: For \(1\)-Dimensional Euclidean Topological Space, Set of Upper Bounded Open Intervals and Lower Bounded Open Intervals Is Subbasis <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, set of upper bounded open intervals and lower bounded open intervals is subbasis Topics About: topological space The table of contents of this article Starting Context Target Contexthttps://thebiasplanet.blogspot.com/2026/05/for1dimensionaleuclideantopologicalspacesetofupperboundedopenintervalsandlowerboundedopenintervalsissubbasis.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7027150587072959731Mon, 04 May 2026 11:33:00 +00002026-05-04T20:33:21.161+09:00Definitions and Propositions1760: For \(2\) Distinct Non-Negative Real Numbers and Natural Number Larger than \(1\), There Are 2nd and 3rd Natural Numbers s.t. 3rd Number Divided by Number to Power of 2nd Number Is Exactly Between Real Numbers <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that for \(2\) distinct non-negative real numbers and natural number larger than \(1\), there are 2nd and 3rd natural numbers s.t. 3rd number divided by number to power of 2nd number is exactly between real numbers Topics About: set The https://thebiasplanet.blogspot.com/2026/05/for2distinctnonnegativerealnumbersandnaturalnumberlargerthan1thereare2ndand3rdnaturalnumbersst3rdnumberdividedbynumbertopowerof2ndnumberisexactlybetweenrealnumbers.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8577238692690655945Sun, 26 Apr 2026 14:55:00 +00002026-05-04T20:31:44.595+09:00Definitions and Propositions1759: 2nd-Countable Completely Regular Topological Space Is Metrizable (Urysohn Metrization Theorem) <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of that 2nd-countable completely regular topological space is metrizable (Urysohn metrization theorem) Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body 1: Structured https://thebiasplanet.blogspot.com/2026/04/2ndcountablecompletelyregulartopologicalspaceismetrizableurysohnmetrizationtheorem.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5608902565568042318Sun, 26 Apr 2026 14:54:00 +00002026-04-26T23:54:05.002+09:00Definitions and Propositions1758: Metrizable Topological Space <The previous article in this series | The table of contents of this series | The next article in this series> definition of metrizable 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 topology https://thebiasplanet.blogspot.com/2026/04/metrizabletopologicalspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1706976770649864311Sun, 26 Apr 2026 14:52:00 +00002026-04-26T23:52:43.147+09:00Definitions and Propositions1757: For 2nd-Countable Completely Regular Topological Space, There Is Countable Set of Continuous Maps into Closed Unit Interval s.t. for Each Point and Each Closed Subset That Does Not Contain Point, There Is Element of Set That Is \(0\) over Open Neighborhood of Point and Is \(1\) over Closed 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 2nd-countable completely regular topological space, there is countable set of continuous maps into closed unit interval s.t. for each point and each closed subset that does not contain point, there is element of set that is \(0\) over open https://thebiasplanet.blogspot.com/2026/04/for2ndcountablecompletelyregulartopologicalspacethereiscountablesetofcontinuousmapsintoclosedunitintervalstforeachpointandeachclosedsubsetthatdoesnotcontainpointthereiselementofsetthatis0overopenneighborhoodofpointandis1overclosedsubset.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7066872522722101816Sun, 26 Apr 2026 14:51:00 +00002026-04-26T23:51:19.253+09:00Definitions and Propositions1756: For Sequence of Metric Spaces with Induced Topologies, This Metric for Product Set Induces Product Topology <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 of metric spaces with induced topologies, this metric for product set induces product topology Topics About: topological space About: metric space The table of contents of this article Starting Context Targethttps://thebiasplanet.blogspot.com/2026/04/forsequenceofmetricspaceswithinducedtopologiesthismetricforproductsetinducesproducttopology.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-4988297505177302908Sun, 26 Apr 2026 14:50:00 +00002026-04-26T23:50:01.439+09:00Definitions and Propositions1755: For Metric Space with Induced Topology and Positive Real Number, Distance as Minimum of Original Distance and Number Is Metric and Induces Original Topology <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 with induced topology and positive real number, distance as minimum of original distance and number is metric and induces original topology Topics About: topological space About: metric space The table of https://thebiasplanet.blogspot.com/2026/04/formetricspacewithinducedtopologyandpositiverealnumberdistanceasminimumoforiginaldistanceandnumberismetricandinducesoriginaltopology.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-469000370923037682Sun, 26 Apr 2026 14:48:00 +00002026-04-26T23:48:35.322+09:00Definitions and Propositions1754: Totally Bounded Subset of Metric Space <The previous article in this series | The table of contents of this series | The next article in this series> definition of totally bounded subset of metric space Topics About: metric 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 https://thebiasplanet.blogspot.com/2026/04/totallyboundedsubsetofmetricspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-8663361670881529462Sun, 26 Apr 2026 14:47:00 +00002026-04-26T23:47:16.233+09:00Definitions and Propositions1753: For Homeomorphism from Metric Space with Induced Topology, Codomain Topology Is Induced by Metric Induced by Homeomorphism <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of for homeomorphism from metric space with induced topology, codomain topology is induced by metric induced by homeomorphism Topics About: topological space About: metric space The table of contents of this article Starting https://thebiasplanet.blogspot.com/2026/04/forhomeomorphismfrommetricspacewithinducedtopologycodomaintopologyisinducedbymetricinducedbyhomeomorphism.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7024929957731800175Sun, 26 Apr 2026 14:45:00 +00002026-04-26T23:45:50.958+09:00Definitions and Propositions1752: Completely Regular Topological Space <The previous article in this series | The table of contents of this series | The next article in this series> definition of completely regular 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 https://thebiasplanet.blogspot.com/2026/04/completelyregulartopologicalspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-7603619818677156421Sun, 26 Apr 2026 14:44:00 +00002026-04-26T23:44:16.128+09:00Definitions and Propositions1751: For Map from Set into Measurable Space, Smallest \(\sigma\)-Algebra of Domain That Makes Map Measurable Is Set of Preimages of Measurable 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 map from set into measurable space, smallest \(\sigma\)-algebra of domain that makes map measurable is set of preimages of measurable subsets Topics About: measurable space The table of contents of this article Starting Context https://thebiasplanet.blogspot.com/2026/04/formapfromsetintomeasurablespacesmallestsigmaalgebraofdomainthatmakesmapmeasurableissetofpreimagesofmeasurablesubsets.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1218903370001788249Sun, 26 Apr 2026 14:42:00 +00002026-04-26T23:42:51.233+09:00Definitions and Propositions1750: Topological Space Is Hausdorff iff Its Diagonal 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 topological space is Hausdorff iff its diagonal is closed 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/topologicalspaceishausdorffiffitsdiagonalisclosed.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-496054030318834000Sun, 26 Apr 2026 14:41:00 +00002026-04-26T23:41:29.923+09:00Definitions and Propositions1749: Product of Regular Topological Spaces 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 product of regular topological spaces 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 Thehttps://thebiasplanet.blogspot.com/2026/04/productofregulartopologicalspacesisregular.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6571171398887875900Sun, 26 Apr 2026 14:40:00 +00002026-04-26T23:40:06.973+09:00Definitions and Propositions1748: For Metric Space with Induced Topology, Points Sequence Converges to Point as on Metric Space iff It Converges to Point as on Topological Space <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 with induced topology, points sequence converges to point as on metric space iff it converges to point as on topological space Topics About: metric space About: topological space The table of contents of https://thebiasplanet.blogspot.com/2026/04/formetricspacewithinducedtopologypointssequenceconvergestopointasonmetricspaceiffitconvergestopointasontopologicalspace.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-6942189154962668895Sun, 26 Apr 2026 14:38:00 +00002026-04-26T23:38:44.139+09:00Definitions and Propositions1747: Topological Space Is Regular iff for Each Point, 1-Point Subset Is Closed and Set of Closed Neighborhoods of Point Is Neighborhoods Basis 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 topological space is regular iff for each point, 1-point subset is closed and set of closed neighborhoods of point is neighborhoods basis at point Topics About: topological space The table of contents of this article Starting Contexthttps://thebiasplanet.blogspot.com/2026/04/topologicalspaceisregulariffforeachpoint1pointsubsetisclosedandsetofclosedneighborhoodsofpointisneighborhoodsbasisatpoint.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-3342155569634505522Sun, 26 Apr 2026 14:37:00 +00002026-04-26T23:37:21.138+09:00Definitions and Propositions1746: Map from Topological Space into Product Topological Space Is Continuous iff Each Component Map 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 map from topological space into product topological space is continuous iff each component map is continuous Topics About: topological space The table of contents of this article Starting Context Target Context Orientation Main Body https://thebiasplanet.blogspot.com/2026/04/mapfromtopologicalspaceintoproducttopologicalspaceiscontinuousiffeachcomponentmapiscontinuous.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-5182398154173296823Sun, 19 Apr 2026 12:55:00 +00002026-04-26T23:35:44.898+09:00Definitions and Propositions1745: Subset of Euclidean Metric Topological Space Is Compact iff Subset Is Closed and Bounded (Heine-Borel Theorem) <The previous article in this series | The table of contents of this series | The next article in this series> description/proof of subset of Euclidean metric topological space is compact iff subset is closed and bounded (Heine-Borel theorem) Topics About: topological space About: metric space The table of contents of this article Starting Context Target https://thebiasplanet.blogspot.com/2026/04/subsetofeuclideanmetrictopologicalspaceiscompactiffsubsetisclosedandbounded.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1159025943656344004Sun, 19 Apr 2026 12:53:00 +00002026-04-19T21:53:34.762+09:00Definitions and Propositions1744: For Product Topological Space and Constituent s.t. Other Constituents Are Compact, Projection onto Constituent 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 for product topological space and constituent s.t. other constituents are compact, projection onto constituent is closed Topics About: topological space The table of contents of this article Starting Context Target Context https://thebiasplanet.blogspot.com/2026/04/forproducttopologicalspaceandconstituentstotherconstituentsarecompactprojectionontoconstituentisclosed.htmlnoreply@blogger.com (Unknown)tag:blogger.com,1999:blog-922111353248289998.post-1149810786614808411Sun, 19 Apr 2026 12:51:00 +00002026-04-19T21:51:15.872+09:00Definitions and Propositions1743: For Product Topological Space and Neighborhood of Point, There Is Open Neighborhood of Point Contained in Neighborhood as Product of Some Open Neighborhoods of Components of 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 product topological space and neighborhood of point, there is open neighborhood of point contained in neighborhood as product of some open neighborhoods of components of point Topics About: topological space The table of contents https://thebiasplanet.blogspot.com/2026/04/forproducttopologicalspaceandneighborhoodofpointthereisopenneighborhoodofpointcontainedinneighborhoodasproductofsomeopenneighborhoodsofcomponentsofpoint.htmlnoreply@blogger.com (Unknown)