September 9th, 2015, 11:34 AM  #11 
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 
You gave an example (actually, several examples) of a "topology" which is not a topology. Country Boy and I explained why those objects are not, in fact, topologies. You seem to have trouble understanding that.

September 9th, 2015, 01:50 PM  #12  
Global Moderator Joined: May 2007 Posts: 6,807 Thanks: 717  Quote:
 
September 10th, 2015, 04:28 AM  #13 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126  Topology and Topological Space
From wiki, General Topology " Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space." The definition (topology) given in Post #2, also given in this wiki article, is a particular, very abstract, definition of an open set, making it a particular topological space. I intuited the definition (same) in the OP from various discussions of topology in textbooks, which never defined topology but clearly the core concept was open set. As luck would have it (in a sense), I stumbled across this wiki article last night. EDIT: The Example topologies (definitions) in OP are topological spaces. Last edited by zylo; September 10th, 2015 at 04:44 AM. 
September 11th, 2015, 12:25 PM  #14  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126  Quote:
If you identify the subsets of S as "open sets" then it is a (particular) topology, and the combination of S and the topology is a topologicaj space (ref wiki above). It may be you claim to give THE abstract definition of topology in which "open set" is dispensed with or is just a meaningless symbol for subset, and a particular definition of subset of S satisfying 1), 2), 3), and 4), along with S, is a topological space. I personally can't think along those lines. I can't memorize symbols, let alone make something meaningful out of them.  
September 14th, 2015, 07:35 AM  #15 
Senior Member Joined: Feb 2012 Posts: 144 Thanks: 16  
September 14th, 2015, 08:13 AM  #16 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126  
October 21st, 2015, 07:59 AM  #17  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126  Quote:
Topology is a family F of subsets of a set X, including X, with Union and Intersection. X plus F is a topological space. Topological Space, Example 1 X: {a,b,c} F: {a},{b},{c},{a,b},{a,c},{b,c},{a,b,c} Union and Intersection, Yes Topological Space, Example 2 X: {a,b,c} F: {a}, {a,b,c} Union and Intersection, Yes Example 3 X: {a,b,c} F: {a},{b},{a,b,c} Union, No. Intersection, Yes I haven't found a geometric definition.  
October 21st, 2015, 09:20 AM  #18 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 
I'm always fine tuning to get the most information with the least memorization. Topology is a family F of subsets of a set X, including X, with all Unions and finite Intersections. X plus F is a topological space. "all" and "finite" are worth carrying. 

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