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)


 * That looks comprehensive and methodical, great work! One small suggestion I would like to make: to give a bit more descriptive names for the corollaries, e.g. "Corollary 1: Pasting Lemma for Continuous Mappings on Open Sets". That way, we emphasize the independent value of the corollaries more, and additionally it will be more intuitive to look for and find their redirects. Other than that, I would say that it is fit for implementation. &mdash; Lord_Farin (talk) 08:15, 5 June 2022 (UTC)


 * As for corollaries, I would even go so far as to not have them directly as corollaries of a result, unless otherwise they would genuinely have a name that makes them horribly unwieldy.
 * Hence instead of calling the page Pasting Lemma/Corollary 1, actually call it Pasting Lemma for Pair of Continuous Mappings on Closed Sets instead.
 * You can of course then use Pasting Lemma/Corollary 1 as a redirect as a page transclusion from Pasting Lemma, but otherwise leave the result itself as a "standalone" page.


 * The advantage of this approach is that if a further proof is implemented that does not directly use Pasting Lemma as a result, but approaches it from a completely different direction, it does not then appear to be "misnamed". ("How can you call xyz a corollary of abc if its proof doesn't use abc anywhere?" We have had at least one discussion on this topic in the past).


 * As I say, I have used the technique of calling a page "corollary" in the past particularly if otherwise the name would be extremely hard to craft.


 * There are anomalies, like Sum of Squares of Sine and Cosine/Corollary 1 and so on, which do indeed look a bit silly. --prime mover (talk) 10:38, 5 June 2022 (UTC)