Fdtxjr

What is it called when someone votes for an option that's not their first choice?

Is VPN a layer 3 concept?

What are the rules for concealing thieves' tools (or items in general)?

Nested Dynamic SOQL Query

Which partition to make active?

How do researchers send unsolicited emails asking for feedback on their works?

How to test the sharpness of a knife?

What kind of footwear is suitable for walking in micro gravity environment?

How can an organ that provides biological immortality be unable to regenerate?

Do people actually use the word "kaputt" in conversation?

is this saw blade faulty?

Did Nintendo change its mind about 68000 SNES?

What are the consequences of changing the number of hours in a day?

How to balance a monster modification (zombie)?

Does fire aspect on a sword, destroy mob drops?

Have any astronauts/cosmonauts died in space?

UK Tourist Visa- Enquiry

Why are there no stars visible in cislunar space?

When did hardware antialiasing start being available?

Single word to change groups

What will the Frenchman say?

Why is indicated airspeed rather than ground speed used during the takeoff roll?

Is xar preinstalled on macOS?

Unfrosted light bulb

Is $mathbb R^J$ normal in the box topology when $J$ is uncountable?

“Proof” that $mathbbR^J$ is not normal when $J$ is uncountableHow to prove that $mathbb R^omega$ with the box topology is completely regularcontinuity box topologyConvergence in the Box Topology.Normal spaces in box and uniform topologyWhat are the components and path components of $mathbbR^omega$ in the product, uniform, and box topologies?$mathbb R^omega$ in the box topology is not first countable.Uncountability of $mathbbR^I$ if $I$ is uncountableconnected components in box topologyMetrizability of infinite product space under box topologyIs $mathbbR^omega$ endowed with the box topology completely normal (or hereditarily normal)?

4

$begingroup$

Question:

Is $mathbb R^J$ normal in the box topology when $J$ is uncountable?

I know $mathbb R^J$ is not normal in the product topology, see "Proof" that $mathbbR^J$ is not normal when $J$ is uncountable ;

I also know $mathbb R^omega$ is normal in the box topology assuming the continuum hypothesis, see Is it still an open problem whether $mathbb R^omega$ is normal in the box topology?.

That's the motivation for this problem. Unfortunately, the above two theorems don't imply anything about the normality of $mathbb R^J$. Any hint would be appreciated.

general-topology separation-axioms box-topology

share|cite|improve this question

edited Mar 13 at 11:20

YuiTo Cheng

asked Mar 13 at 9:54

YuiTo ChengYuiTo Cheng

2,0592637

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  What do you mean by "Box topology" ? It's not a standard naming...
  $endgroup$
  – Jean Marie
  Mar 13 at 12:12
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @JeanMarie Maybe box product sounds better?
  $endgroup$
  – YuiTo Cheng
  Mar 13 at 12:13
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  A closed subspace of a normal space is normal. $Bbb R^omega$ is homeomorphic to a closed subspace of $Bbb R^k$ if $k$ is uncountable.
  $endgroup$
  – DanielWainfleet
  Mar 13 at 14:49
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @DanielWainfleet So you are suggesting this problem is likely to be open, right?
  $endgroup$
  – YuiTo Cheng
  Mar 13 at 14:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If the countable product isn't normal (which is open) then so would the higher powers be.
  $endgroup$
  – Henno Brandsma
  Mar 13 at 16:49
  

  
  
  
  

 | 
show 2 more comments

4

$begingroup$

Question:

Is $mathbb R^J$ normal in the box topology when $J$ is uncountable?

I know $mathbb R^J$ is not normal in the product topology, see "Proof" that $mathbbR^J$ is not normal when $J$ is uncountable ;

I also know $mathbb R^omega$ is normal in the box topology assuming the continuum hypothesis, see Is it still an open problem whether $mathbb R^omega$ is normal in the box topology?.

That's the motivation for this problem. Unfortunately, the above two theorems don't imply anything about the normality of $mathbb R^J$. Any hint would be appreciated.

general-topology separation-axioms box-topology

share|cite|improve this question

edited Mar 13 at 11:20

YuiTo Cheng

asked Mar 13 at 9:54

YuiTo ChengYuiTo Cheng

2,0592637

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  What do you mean by "Box topology" ? It's not a standard naming...
  $endgroup$
  – Jean Marie
  Mar 13 at 12:12
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @JeanMarie Maybe box product sounds better?
  $endgroup$
  – YuiTo Cheng
  Mar 13 at 12:13
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  A closed subspace of a normal space is normal. $Bbb R^omega$ is homeomorphic to a closed subspace of $Bbb R^k$ if $k$ is uncountable.
  $endgroup$
  – DanielWainfleet
  Mar 13 at 14:49
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @DanielWainfleet So you are suggesting this problem is likely to be open, right?
  $endgroup$
  – YuiTo Cheng
  Mar 13 at 14:53
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If the countable product isn't normal (which is open) then so would the higher powers be.
  $endgroup$
  – Henno Brandsma
  Mar 13 at 16:49
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

Question:

Is $mathbb R^J$ normal in the box topology when $J$ is uncountable?

I know $mathbb R^J$ is not normal in the product topology, see "Proof" that $mathbbR^J$ is not normal when $J$ is uncountable ;

I also know $mathbb R^omega$ is normal in the box topology assuming the continuum hypothesis, see Is it still an open problem whether $mathbb R^omega$ is normal in the box topology?.

That's the motivation for this problem. Unfortunately, the above two theorems don't imply anything about the normality of $mathbb R^J$. Any hint would be appreciated.

general-topology separation-axioms box-topology

share|cite|improve this question

edited Mar 13 at 11:20

YuiTo Cheng

asked Mar 13 at 9:54

YuiTo ChengYuiTo Cheng

2,0592637

$endgroup$

share|cite|improve this question

edited Mar 13 at 11:20

YuiTo Cheng

asked Mar 13 at 9:54

YuiTo ChengYuiTo Cheng

2,0592637

share|cite|improve this question

edited Mar 13 at 11:20

YuiTo Cheng

asked Mar 13 at 9:54

YuiTo ChengYuiTo Cheng

2,0592637

asked Mar 13 at 9:54

YuiTo ChengYuiTo Cheng

2,0592637

asked Mar 13 at 9:54

YuiTo ChengYuiTo Cheng

2,0592637