Dgdrxrt

How does the UK government determine the size of a mandate?

Arithmetic mean geometric mean inequality unclear

What is the best translation for "slot" in the context of multiplayer video games?

Energy of the particles in the particle accelerator

Is the destination of a commercial flight important for the pilot?

How do scammers retract money, while you can’t?

How do I go from 300 unfinished/half written blog posts, to published posts?

Is there a korbon needed for conversion?

Pre-amplifier input protection

Is expanding the research of a group into machine learning as a PhD student risky?

Failed to fetch jessie backports repository

Customer Requests (Sometimes) Drive Me Bonkers!

Inappropriate reference requests from Journal reviewers

Hostile work environment after whistle-blowing on coworker and our boss. What do I do?

How to Reset Passwords on Multiple Websites Easily?

Would this custom Sorcerer variant that can only learn any verbal-component-only spell be unbalanced?

System.debug(JSON.Serialize(o)) Not longer shows full string

Applicability of Single Responsibility Principle

How does it work when somebody invests in my business?

Large drywall patch supports

Unreliable Magic - Is it worth it?

How can I get through very long and very dry, but also very useful technical documents when learning a new tool?

Why not increase contact surface when reentering the atmosphere?

Is a stroke of luck acceptable after a series of unfavorable events?

Does the box topology have a universal property?

Examples on product topology $ gg $ box topology?Disjoint Union Topology universal property as instance of final topology's universal propertyuniversal property in quotient topologyTheorem 19.4 in Munkres' TOPOLOGY, 2nd ed: Does the converse also hold?Universal property of topology of uniform convergenceGenerating family for box topologyuniversal property of measurable spacesUniversal Property of Disjoint Sum topologyDefining a universal propertyBox topology defines a topological vector space?

11

2

$begingroup$

Given a set of topological spaces $X_alpha$, there are two main topologies we can give to the Cartesian product $Pi_alpha X_alpha$: the product topology and the box topology. The product topology has the following universal property: given a topological space $Y$ and a family $f_alpha$ of continuous maps from $Y$ to each $X_alpha$, there exists a continuous map from $Y$ to $Pi_alpha X_alpha$. Now the box topology does not have this universal property, but my question is, does it have some other universal property?

On a related note, does there exist some category whose objects are topological spaces and whose morphisms are something other than continuous maps, such that the Cartesian product endowed with the box topology is the correct product object in that category?

general-topology category-theory product-space universal-property box-topology

share|cite|improve this question

asked Feb 1 at 4:18

Keshav SrinivasanKeshav Srinivasan

2,36921446

$endgroup$

This question has an open bounty worth +50
reputation from Keshav Srinivasan ending ending at 2019-04-03 01:14:48Z">in 6 days.

This question has not received enough attention.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Here's maybe a starting point: A universal property is usually defined in terms of (co)limits. Now the forgetful functor from the set has both left and right adjoints (the two simplest choices for topologies: trivial and discrete topologies for each set). Right adjoints preserve limits and left adjoints preserve colimits. Thus a universal property of the box topology would still have to be a universal property of the underlying set (and the underlying sets and functions of whatever diagram you chose to pose the universal problem that the (co)limit solves).
  $endgroup$
  – ffffforall
  Feb 1 at 5:02
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Here's a thought : the box topology actually behaves somewhat like a coproduct. Indeed, for $S$ the Sierpinski space, maps from the box to $S$ correspond to families of maps to $S$. More generally, it will be easier to find maps out of the box than into it (if there are infinitely many spaces).
  $endgroup$
  – Max
  22 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Here's a claim that the box topology is the product among maps "such that locally almost all members are constant". books.google.com/books?id=l-XxBwAAQBAJ&pg=PA265
  $endgroup$
  – Chris Culter
  1 hour ago
  

  
  
  
  

add a comment | 

11

2

$begingroup$

Given a set of topological spaces $X_alpha$, there are two main topologies we can give to the Cartesian product $Pi_alpha X_alpha$: the product topology and the box topology. The product topology has the following universal property: given a topological space $Y$ and a family $f_alpha$ of continuous maps from $Y$ to each $X_alpha$, there exists a continuous map from $Y$ to $Pi_alpha X_alpha$. Now the box topology does not have this universal property, but my question is, does it have some other universal property?

On a related note, does there exist some category whose objects are topological spaces and whose morphisms are something other than continuous maps, such that the Cartesian product endowed with the box topology is the correct product object in that category?

general-topology category-theory product-space universal-property box-topology

share|cite|improve this question

asked Feb 1 at 4:18

Keshav SrinivasanKeshav Srinivasan

2,36921446

$endgroup$

This question has an open bounty worth +50
reputation from Keshav Srinivasan ending ending at 2019-04-03 01:14:48Z">in 6 days.

This question has not received enough attention.

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Here's maybe a starting point: A universal property is usually defined in terms of (co)limits. Now the forgetful functor from the set has both left and right adjoints (the two simplest choices for topologies: trivial and discrete topologies for each set). Right adjoints preserve limits and left adjoints preserve colimits. Thus a universal property of the box topology would still have to be a universal property of the underlying set (and the underlying sets and functions of whatever diagram you chose to pose the universal problem that the (co)limit solves).
  $endgroup$
  – ffffforall
  Feb 1 at 5:02
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Here's a thought : the box topology actually behaves somewhat like a coproduct. Indeed, for $S$ the Sierpinski space, maps from the box to $S$ correspond to families of maps to $S$. More generally, it will be easier to find maps out of the box than into it (if there are infinitely many spaces).
  $endgroup$
  – Max
  22 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Here's a claim that the box topology is the product among maps "such that locally almost all members are constant". books.google.com/books?id=l-XxBwAAQBAJ&pg=PA265
  $endgroup$
  – Chris Culter
  1 hour ago
  

  
  
  
  

add a comment | 

$begingroup$

Given a set of topological spaces $X_alpha$, there are two main topologies we can give to the Cartesian product $Pi_alpha X_alpha$: the product topology and the box topology. The product topology has the following universal property: given a topological space $Y$ and a family $f_alpha$ of continuous maps from $Y$ to each $X_alpha$, there exists a continuous map from $Y$ to $Pi_alpha X_alpha$. Now the box topology does not have this universal property, but my question is, does it have some other universal property?

On a related note, does there exist some category whose objects are topological spaces and whose morphisms are something other than continuous maps, such that the Cartesian product endowed with the box topology is the correct product object in that category?

general-topology category-theory product-space universal-property box-topology

share|cite|improve this question

asked Feb 1 at 4:18

Keshav SrinivasanKeshav Srinivasan

2,36921446

$endgroup$

share|cite|improve this question

asked Feb 1 at 4:18

Keshav SrinivasanKeshav Srinivasan

2,36921446

share|cite|improve this question

asked Feb 1 at 4:18

Keshav SrinivasanKeshav Srinivasan

2,36921446

asked Feb 1 at 4:18

Keshav SrinivasanKeshav Srinivasan

2,36921446

asked Feb 1 at 4:18

Keshav SrinivasanKeshav Srinivasan

2,36921446