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What's the difference between different approaches to demonstrating closedness (etc.) sets?

Heine-Borel - Finite Intersection equivalence in subspacesCompact subset of a Hausdorff space is closed.What is the difference between open intervals and standard topology on $mathbbR$How to prove any closed set in $mathbbR$ is $G_delta$Limit point definition of a closed set, versus complement of open set definition?Complements on the product topologyShowing a set is closed and show a set is open(if and only if)Consider the sequence $x_n = (7 + (-1)^n/n, -5) inmathbbR^2$. What are the sets of limit points in different topologies?Please, where can I find some good worked examples of compact spaces?The complement of an open set?

0

$begingroup$

What's the difference between different approaches to demonstrating closedness (etc.) sets?

What confuses me that sometimes I for example find showing

complement is open $implies$ the set is closed

easier than some

"take a sequence of elements and show that limit is not in the set"

---

The problem with this latter approach is that I find that depending on the set it could be difficult to construct such sequence.

---

Or would you say that these are equivalent in terms of "difficulty"?

general-topology

share|cite|improve this question

edited Mar 22 at 23:57

mavavilj

asked Mar 22 at 15:20

mavaviljmavavilj

2,85911138

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  It's hard to know how to answer your question. Mathematical methods apply differently to different mathematical problems. "Closedness" might be demonstrated one way in one problem and a different way in a different problem. Learning the correct way to demonstrate it in a particular problem is part of the job of the mathematician. If you have a question about doing this in a particular problem that's one thing, but as the question is currently posed I don't think it is answerable.
  $endgroup$
  – Lee Mosher
  Mar 22 at 15:36
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Put a concrete example where "could be difficult to construct such sequence".
  $endgroup$
  – Martín-Blas Pérez Pinilla
  Mar 22 at 16:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Martín-BlasPérezPinilla $ n : n in mathbbZ_+$ is closed, but other than noticing that its complement is union of open intervals $(n,n+1)$, then how would one prove that some sequence would not have a limit in this set? I guess one could show that the sequence difference cannot be less than one?
  $endgroup$
  – mavavilj
  Mar 22 at 17:59
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Right. Using that the space is discrete is the essential fact.
  $endgroup$
  – Martín-Blas Pérez Pinilla
  Mar 22 at 21:12
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That "take a sequence of elements" stuff does not always hold.
  $endgroup$
  – William Elliot
  Mar 22 at 23:11
  

  
  
  
  

 | 
show 2 more comments

0

$begingroup$

What's the difference between different approaches to demonstrating closedness (etc.) sets?

What confuses me that sometimes I for example find showing

complement is open $implies$ the set is closed

easier than some

"take a sequence of elements and show that limit is not in the set"

---

The problem with this latter approach is that I find that depending on the set it could be difficult to construct such sequence.

---

Or would you say that these are equivalent in terms of "difficulty"?

general-topology

share|cite|improve this question

edited Mar 22 at 23:57

mavavilj

asked Mar 22 at 15:20

mavaviljmavavilj

2,85911138

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  It's hard to know how to answer your question. Mathematical methods apply differently to different mathematical problems. "Closedness" might be demonstrated one way in one problem and a different way in a different problem. Learning the correct way to demonstrate it in a particular problem is part of the job of the mathematician. If you have a question about doing this in a particular problem that's one thing, but as the question is currently posed I don't think it is answerable.
  $endgroup$
  – Lee Mosher
  Mar 22 at 15:36
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Put a concrete example where "could be difficult to construct such sequence".
  $endgroup$
  – Martín-Blas Pérez Pinilla
  Mar 22 at 16:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Martín-BlasPérezPinilla $ n : n in mathbbZ_+$ is closed, but other than noticing that its complement is union of open intervals $(n,n+1)$, then how would one prove that some sequence would not have a limit in this set? I guess one could show that the sequence difference cannot be less than one?
  $endgroup$
  – mavavilj
  Mar 22 at 17:59
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Right. Using that the space is discrete is the essential fact.
  $endgroup$
  – Martín-Blas Pérez Pinilla
  Mar 22 at 21:12
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That "take a sequence of elements" stuff does not always hold.
  $endgroup$
  – William Elliot
  Mar 22 at 23:11
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

What's the difference between different approaches to demonstrating closedness (etc.) sets?

What confuses me that sometimes I for example find showing

complement is open $implies$ the set is closed

easier than some

"take a sequence of elements and show that limit is not in the set"

---

The problem with this latter approach is that I find that depending on the set it could be difficult to construct such sequence.

---

Or would you say that these are equivalent in terms of "difficulty"?

general-topology

share|cite|improve this question

edited Mar 22 at 23:57

mavavilj

asked Mar 22 at 15:20

mavaviljmavavilj

2,85911138

$endgroup$

share|cite|improve this question

edited Mar 22 at 23:57

mavavilj

asked Mar 22 at 15:20

mavaviljmavavilj

2,85911138

share|cite|improve this question

edited Mar 22 at 23:57

mavavilj

asked Mar 22 at 15:20

mavaviljmavavilj

2,85911138

asked Mar 22 at 15:20

mavaviljmavavilj

2,85911138

asked Mar 22 at 15:20

mavaviljmavavilj

2,85911138