Talk:Pasting Lemma

Pages to be merge
I've taken a stab at refactoring the following pages:
 * Pasting Lemma
 * Pasting Lemma/Corollary 1::Pasting Lemma/Corollary 2
 * Continuity from Union of Restrictions
 * Continuous Mapping on Finite Union of Closed Sets
 * Continuous Mapping on Union of Open Sets

New proposed pages
I have refactored all of these pages under a top level page to be called Pasting Lemma.

Beneath this top level page are two subpages: one for open sets; the other for closed sets.

These pages state the lemma from a top-down perspective: a function is continuous when restricted to open/closed sets.

Below each of these subpages are two corollaries. One of the corollaries presents the pasting lemma from a bottom-up persepective: given a family of continuous functions the union function is continuous. The other corollary presents the pasting lemma from a bottom-up perspective given a pair of continuous functions.

See these pages for proposed changes:


 * User:Leigh.Samphier/Topology/Pasting Lemma


 * User:Leigh.Samphier/Topology/Pasting Lemma/Union of Open Sets


 * User:Leigh.Samphier/Topology/Pasting Lemma/Union of Open Sets/Corollary 1


 * User:Leigh.Samphier/Topology/Pasting Lemma/Union of Open Sets/Corollary 2


 * User:Leigh.Samphier/Topology/Pasting Lemma/Finite Union of Closed Sets


 * User:Leigh.Samphier/Topology/Pasting Lemma/Finite Union of Closed Sets/Corollary 1


 * User:Leigh.Samphier/Topology/Pasting Lemma/Finite Union of Closed Sets/Corollary 2


 * User:Leigh.Samphier/Topology/Pasting Lemma/Counterexample of Infinite Union of Closed Sets

What replaces what?
Pasting Lemma replaced by User:Leigh.Samphier/Topology/Pasting Lemma/Union of Open Sets/Corollary 2

Pasting Lemma/Corollary 1 replaced by User:Leigh.Samphier/Topology/Pasting Lemma/Finite Union of Closed Sets/Corollary 2

Pasting Lemma/Corollary 2 replaced by User:Leigh.Samphier/Topology/Pasting Lemma/Union of Open Sets

Continuity from Union of Restrictions replaced by User:Leigh.Samphier/Topology/Pasting Lemma

Continuous Mapping on Finite Union of Closed Sets replaced by User:Leigh.Samphier/Topology/Pasting Lemma/Finite Union of Closed Sets

Continuous Mapping on Union of Open Sets replaced by User:Leigh.Samphier/Topology/Pasting Lemma/Union of Open Sets

Proposed redirect pages
The following redirect pages will be created or changed.


 * Pasting Lemma for Union of Open Sets => Pasting Lemma/Union of Open Sets


 * Continuous Mapping on Union of Open Sets => Pasting Lemma/Union of Open Sets


 * Pasting Lemma for Continuous Mappings on Open Sets => Pasting Lemma/Union of Open Sets/Corollary 1


 * Pasting Lemma for Pair of Continuous Mappings on Open Sets => Pasting Lemma/Union of Open Sets/Corollary 2


 * Pasting Lemma for Finite Union of Closed Sets => Pasting Lemma/Finite Union of Closed Sets


 * Continuous Mapping on Finite Union of Closed Sets => Pasting Lemma/Finite Union of Closed Sets


 * Pasting Lemma for Continuous Mappings on Closed Sets => Pasting Lemma/Finite Union of Closed Sets/Corollary 1


 * Pasting Lemma for Pair of Continuous Mappings on Closed Sets => Pasting Lemma/Finite Union of Closed Sets/Corollary 2

Changes to existing links

 * Pasting Lemma
 * Leave these pages linked to Pasting Lemma
 * Continuity from Union of Restrictions
 * Continuous Mapping on Finite Union of Closed Sets
 * Power Function to Rational Power permits Unique Continuous Extension
 * User:Keith.U


 * Pasting Lemma/Corollary 1
 * Change link to Pasting Lemma for Pair of Continuous Mappings on Closed Sets in
 * User:Keith.U


 * Pasting Lemma/Corollary 2
 * Change link to Pasting Lemma for Pair of Continuous Mappings on Open Sets in
 * User:Keith.U


 * Continuity from Union of Restrictions
 * Change link in these pages
 * Continuous Mapping of Separation change to Pasting Lemma for Union of Open Sets
 * Joining Paths makes Another Path change to Pasting Lemma for Finite Union of Closed Sets
 * Projection from Product Topology is Open change to Pasting Lemma
 * Diameter of Closure of Subset is Diameter of Subset change to Pasting Lemma


 * Continuous Mapping on Finite Union of Closed Sets
 * Leave linked to Continuous Mapping on Finite Union of Closed Sets
 * Relative Homotopy is Equivalence Relation
 * Continuity from Union of Restrictions
 * Trivial Topological Space is Non-Meager
 * Continuous Mapping on Union of Open Sets
 * Homotopic Paths Implies Homotopic Composition
 * Definition:Concatenation (Topology)


 * Continuous Mapping on Union of Open Sets
 * Leave linked to Continuous Mapping on Union of Open Sets
 * Continuity from Union of Restrictions
 * Continuous Mapping on Finite Union of Closed Sets

--Leigh.Samphier (talk) 16:53, 4 June 2022 (UTC)