Dtjytyk

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Is there a way to give a caracterization of a topology using convergent sequences?

Do limits of sequences of sets come from a topology?Convergence/Sequences/Box TopologyThe space with countable complement topology (example 20 in “Counterexamples in topology”)Does $X=[0,omega_1]$ satisfy $S_1(Omega,Omega)$?Is $mathbb{R^2}$ Hausdorff? Give an example of a non-Hausdorff topology on $mathbb{R}$Counterexample to the intersection of compact sets being compact.The isolated point in Hausdorff spaceTopology on the Set of Convergent SequencesGive an example of a topology that is not compact, not connected, and not HausdorffEvery $F_sigma$-set in a paracompact space is paracompact.

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Suppose I want to define a space $mathcal L^1(mathbb R)$ the set of function s.t. $-infty <int_{mathbb R}f<infty $. And now I say that a sequence $(f_n)$ of element of $mathcal L^1(mathbb R)$ converge to a function $fin mathcal L^1$ if $$lim_{nto infty }int_{mathbb R}f_n=int_{mathbb R}f.$$
Is there a way to find a topology from that ? What would be the open of this set ?

---

I know that that my topological space is not really interesting... and that it won't be hausdorff since for example $f_n=boldsymbol 1_{[n,n+1]}$ will have a lot of different limits (for example it will converges to $f_a=boldsymbol 1_{[a,a+1]}$ for all $ainmathbb R$, the we'll have at least an uncountable number of different limit). But I would be very interested if we can find the open of my new $mathcal L^1(mathbb R)$ space.

general-topology

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edited yesterday

Pierre

asked yesterday

PierrePierre

284

New contributor

Pierre is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Should the elements of $mathcal{L}^1(mathbb{R})$ be non-negative a.e.? If not, then the condition $int_{mathbb{R}} f < infty$ would not omit the possibility that $int_{mathbb{R}} f = - infty$.
  $endgroup$
  – rolandcyp
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @rolandcyp: I edited my question.
  $endgroup$
  – Pierre
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You may find this useful. mathoverflow.net/questions/36379/…
  $endgroup$
  – Robert Thingum
  yesterday
  
  
  

  
  
  
  

add a comment | 

1

1

$begingroup$

Suppose I want to define a space $mathcal L^1(mathbb R)$ the set of function s.t. $-infty <int_{mathbb R}f<infty $. And now I say that a sequence $(f_n)$ of element of $mathcal L^1(mathbb R)$ converge to a function $fin mathcal L^1$ if $$lim_{nto infty }int_{mathbb R}f_n=int_{mathbb R}f.$$
Is there a way to find a topology from that ? What would be the open of this set ?

---

I know that that my topological space is not really interesting... and that it won't be hausdorff since for example $f_n=boldsymbol 1_{[n,n+1]}$ will have a lot of different limits (for example it will converges to $f_a=boldsymbol 1_{[a,a+1]}$ for all $ainmathbb R$, the we'll have at least an uncountable number of different limit). But I would be very interested if we can find the open of my new $mathcal L^1(mathbb R)$ space.

general-topology

share|cite|improve this question

edited yesterday

Pierre

asked yesterday

PierrePierre

284

New contributor

Pierre is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Should the elements of $mathcal{L}^1(mathbb{R})$ be non-negative a.e.? If not, then the condition $int_{mathbb{R}} f < infty$ would not omit the possibility that $int_{mathbb{R}} f = - infty$.
  $endgroup$
  – rolandcyp
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @rolandcyp: I edited my question.
  $endgroup$
  – Pierre
  yesterday
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You may find this useful. mathoverflow.net/questions/36379/…
  $endgroup$
  – Robert Thingum
  yesterday
  
  
  

  
  
  
  

add a comment | 

$begingroup$

Suppose I want to define a space $mathcal L^1(mathbb R)$ the set of function s.t. $-infty <int_{mathbb R}f<infty $. And now I say that a sequence $(f_n)$ of element of $mathcal L^1(mathbb R)$ converge to a function $fin mathcal L^1$ if $$lim_{nto infty }int_{mathbb R}f_n=int_{mathbb R}f.$$
Is there a way to find a topology from that ? What would be the open of this set ?

---

I know that that my topological space is not really interesting... and that it won't be hausdorff since for example $f_n=boldsymbol 1_{[n,n+1]}$ will have a lot of different limits (for example it will converges to $f_a=boldsymbol 1_{[a,a+1]}$ for all $ainmathbb R$, the we'll have at least an uncountable number of different limit). But I would be very interested if we can find the open of my new $mathcal L^1(mathbb R)$ space.

general-topology

share|cite|improve this question

edited yesterday

Pierre

asked yesterday

PierrePierre

284

New contributor

Pierre is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited yesterday

Pierre

asked yesterday

PierrePierre

284

New contributor

Pierre is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited yesterday

Pierre

asked yesterday

PierrePierre

284

New contributor

Pierre is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked yesterday

PierrePierre

284

New contributor

Pierre is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked yesterday

PierrePierre

284