My Math Forum Topology Definition

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 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
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 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 sub-collection of that collection, the union of all sets in A is also in A. 4) If B is a finite sub-collection of that collection, the intersection of all sets in B is also in B.
This doesn't look right. Assuming C is the collection referred to in the definition. The endings for statements 3 and 4 should be "in C", not "in A" or "in B". In other words any unions (A) or any finite intersections (B) of sets in C is in C.

 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
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 Originally Posted by Country Boy 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 sub-collection of that collection, the union of all sets in A is also in A. 4) If B is a finite sub-collection of that collection, the intersection of all sets in B is also in B.
The above is not a topology because it doesn't define "open set."
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
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 Originally Posted by zylo I can't memorize symbols, let alone make something meaningful out of them.
stay away from mathematics.

September 14th, 2015, 08:13 AM   #16
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 Originally Posted by mehoul stay away from mathematics.
There's more to mathematics than memorizing symbols.

October 21st, 2015, 07:59 AM   #17
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Quote:
 Originally Posted by Country Boy 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 sub-collection of that collection, the union of all sets in A is also in A. 4) If B is a finite sub-collection of that collection, the intersection of all sets in B is also in B.
Yes, it's the definition in Taylor. Sorry I didn't come across it earlier, would have saved a lot of blah-blah on my part. As a personal mnemonic I would use:

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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