September 9th, 2017, 03:17 PM  #11 
Senior Member Joined: Jun 2015 From: England Posts: 890 Thanks: 268 
So why is everybody so determined to use undefined terms instead of the word set? A family of subsets A collection of subsets but never A set of subsets 
September 9th, 2017, 10:33 PM  #12  
Senior Member Joined: Aug 2012 Posts: 2,075 Thanks: 593  Quote:
Note that there is no possible ambiguity with proper classes. By the powerset axiom, the family or collection of subsets of a given set is itself a set. So instead of saying, "Let X be a set of subsets of Y" they say "Let X be a collection ..." I don't think anyone would say, "Let X be a collection of sets," because that IS ambiguous. Is X a set or a proper class? We have no way of knowing. But when the collection in question is a subset of the powerset of some set, we know it's a class and there is no ambiguity. I think I didn't really answer your question. I only agreed that everyone does this. I don't know how it got started. I think "set of sets" sounds a little awkward and "family" doesn't. Last edited by skipjack; September 10th, 2017 at 12:21 AM.  
September 10th, 2017, 01:43 AM  #13 
Senior Member Joined: Jun 2015 From: England Posts: 890 Thanks: 268 
Thank you Maschke. If T is a set then why is is necessary to specify explicitly that T contains the empty set? Surely a more fundamental axiom or (result?) is that every set contains the empty set, therefore T contains it? 
September 10th, 2017, 06:47 AM  #14 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 282 Thanks: 85 Math Focus: Algebraic Number Theory, Arithmetic Geometry  
September 10th, 2017, 09:15 AM  #15  
Senior Member Joined: Jun 2015 From: England Posts: 890 Thanks: 268  Quote:
I meant that the empty set is a subset of every set, not that it is a member of every set.  
September 10th, 2017, 09:34 AM  #16  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 282 Thanks: 85 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Quote:
As you've explained, it's unnecessary to specify that the empty set is a subset of T. Fortunately, the definition does not do this. Last edited by cjem; September 10th, 2017 at 09:37 AM.  
September 10th, 2017, 10:28 AM  #17 
Senior Member Joined: Jun 2015 From: England Posts: 890 Thanks: 268 
Ah clarity, thanks.

October 17th, 2017, 05:08 PM  #18 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
One prefers not to use the phrase "set of sets" because of the difficulty proving that there is such a thing as a result of "Russel's paradox", https://plato.stanford.edu/entries/russellparadox/.

October 17th, 2017, 07:09 PM  #19 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,906 Thanks: 771 Math Focus: Wibbly wobbly timeywimey stuff. 
It seems that I have missed the other thread. @Zylo: As a stand alone thread this is a bit noninformative. Okay, so you've got your definition of a topology (however lax you might be about using the specific terms, but no matter.) So far so good. But what is your point? Dan 

Tags 
definition, topology 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Topology Definition  zylo  Topology  17  October 21st, 2015 10:20 AM 
Problem on product topology/standard topology on R^2.  vercammen  Topology  1  October 19th, 2012 12:06 PM 
Topology  Artus  Topology  5  September 5th, 2012 08:21 AM 
discrete topology, product topology  genoatopologist  Topology  0  December 6th, 2008 11:09 AM 
discrete topology, product topology  Erdos32212  Topology  0  December 2nd, 2008 02:04 PM 