September 7th, 2015, 08:26 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,390 Thanks: 100  Topology Definition
Topology on a set S is the definition of an open set in S. Examples 1) Topology on nspace: A set S in nspace is open if every point in S has a neighborhood in nspace consisting only of points in S. 1a) Topology (relative) on a set S in nspace: A set U in S is open if every point of U has a neighborhood in S consisting only of points in U. 2) A Topology on red and blue marbles in a box. A set S of marbles in the box is open if every marble in S touches only marbles of the same color. 2a) A Topology on the set S of red marbles in the box. A set U in S is open if the red marbles touching U are members of U. Associated with (dependent on) definition of open are closed, complement, boundary, etc. 
September 7th, 2015, 10:57 AM  #2  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,242 Thanks: 885  More precisely, a "topology" for a set S is a collection of subsets of S such that 1) The entire set, S, is in that collection. 2) The empty set is in that collection. 3) If A is a subcollection of that collection, the union of all sets in A is also in A. 4) If B is a finite subcollection of that collection, the intersection of all sets in B is also in B. Quote:
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September 8th, 2015, 06:45 AM  #3  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,390 Thanks: 100  Quote:
S={a,b,S}={a,b,{a,b,S}}={a,b,{a,b,{a,b,S}}}=...... ........... That is why Russell's Paradox is not a paradox: it deals with something that doesn't exist a set that is a member of itself. 2), 3), and 4) are sophistries of set theory irrelevant to fundamental definition of Topology. I believe this deals with your other objections. I have proposed a definition that is concise, precise, easy to remember, and easy to use. You disagree. OK, no hard feelings. We think differently.  
September 8th, 2015, 07:19 AM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Quote:
Also, there are set theories which allow sets to be members of themselves, like the Quine atom Q = {Q}. ZF doesn't, though. Quote:
Charitably, you may have misunderstood the points raised above.  
September 8th, 2015, 11:31 AM  #5  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,390 Thanks: 100  Quote:
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The topic is not axiomatc set theory, which doesn't even derive from a definition of topology. Quote:
Halmos, "Naive Set Theory"; Suppes, "Axiomatic Set Theory" and Quine, "Set Theory and its Logic" don't even mention "topology" (index).  
September 8th, 2015, 03:48 PM  #6 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,242 Thanks: 885 
I'll help you reread it. I said "More precisely, a "topology" for a set S is a collection of subsets of S such that 1) The entire set, S, is in that collection." "S is in that collection", the topology, not in S. Or are you complaining that S is not a subset of S? If so you need to check the definition of "subset"! Being a subset of S is not the same as being a member of S. 
September 9th, 2015, 06:43 AM  #7  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,390 Thanks: 100  Quote:
My gripe with abstraction is that it replaces the insight and origin of the source (usually ignored) with the abstractionist. What is the source? *S={a,b,c} C=[{a},{a,b},{a,c},{b,c},{a,b,c},{}] 3) A=[{a,b},{a,c}], {a,b}U{a,c}={a,b,c}, not in A. 4) B=[{a,b},{a,b,c}], {a,b}I{a,b,c}={a}, not in B So this is not an example of a "topology." Do you have an example? Thanks for response. It is interesting. Last edited by zylo; September 9th, 2015 at 06:48 AM.  
September 9th, 2015, 07:20 AM  #8 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,390 Thanks: 100 
Country Boy You gave a definition of Topological Space which I saw in a neighboring post and googled. Why didn't you say so? Sorry I thanked you. The only thing left standing after all this is the OP definition: "Topology on a set S is the definition of an open set in S." 
September 9th, 2015, 07:41 AM  #9 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
The things that are in a topology are (by definition) open in that topology. But that doesn't mean you can choose any old set to be a topology! You need the properties of the topology to hold: it must contain the whole space and be closed under unions and finite intersections.

September 9th, 2015, 08:18 AM  #10  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,390 Thanks: 100  Quote:
Unfortunately, the discussion got bushwacked by someone introducing topological spaces without saying so. Next: More misrepresentations of what I said. Politics101  

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