My Math Forum A Definition of Topology

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 September 2nd, 2017, 01:17 PM #1 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 A Definition of Topology "Basic concept (building blocks) of Topology: A set and all it's subsets. "X" Topology: A Set and a specific collection of subsets known as "open subsets." (S,s). If "open subsets" = open subsets, you have Euclidean Topology. REF: Neighbourhood of a point Last edited by zylo; September 2nd, 2017 at 01:22 PM. Reason: added REF
 September 2nd, 2017, 04:20 PM #2 Senior Member   Joined: Aug 2017 From: United Kingdom Posts: 307 Thanks: 101 Math Focus: Number Theory, Algebraic Geometry Ah! I understand what you mean by "open sets" and open sets now. It's actually quite a special case when your set $X$ has some obvious metric like Euclidean distance to refer to, so it's a little odd to talk about open sets (in the way you mean them) in the general case. This is probably why I didn't get what you meant before!
September 2nd, 2017, 06:55 PM   #3
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 Originally Posted by zylo "Basic concept (building blocks) of Topology: A set and all it's subsets. "X" Topology: A Set and a specific collection of subsets known as "open subsets." (S,s). If "open subsets" = open subsets, you have Euclidean Topology.
I appreciate that you've moved this discussion to its own thread.

Zylo, You have not provided a definition.

One of the most important attributes of a definition is that it should uniquely characterize the thing it's defining. A definition that's too broad, that describes things that you mean and also things that you don't mean, is no definition at all.

Math is often about finding the right definition. As Michael Spivak says in Calculus on Manifolds, "The theorems should all be easy and the definitions hard."

The history of math can be seen as the search for the right definitions. It took 200 years to find the right definition of limit, from Newton to Cauchy and Weierstrass. People studied polynomials for millennia before Galois understood that the key insight was the definition of a group.

So we should be careful and thoughtful when we make definitions.

Now Zylo you said that a topology is a set along with some collection of its subsets.

I do hope that you can see, without my needing to write down examples, that this is far too broad. It includes things like the usual topology on the reals, which you allude to, but it also includes many collections that are not topologies. You should supply some examples to show that you understand what I'm saying.

So what you say is indeed true about topologies, as far as it goes. But it does not go far enough to serve as a definition of a topology.

And I hope that you can see this.

Last edited by Maschke; September 2nd, 2017 at 07:13 PM.

 September 4th, 2017, 08:16 AM #4 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 It's simply an all-encompassing description for the subject, ei, Calculus: the study of limits. Initial protocols for the internet were very short and broad (Application, Transport, Internet, Link) I mean that modestly, in the sense that it's very simple and easy to remember. And you can flesh it out with easily understood specifics. I can't remember definitions with all the specifics for a particular case. If you start from specifics you lose perspective. It's for the masses. A high school student, even elementary, can understand it: Put blocks in a bucket- the SET. Then take some buckets and label them "Topology 1," "Topology 2," "Topology 3" and ask the child to put various combinations of blocks in each bucket. Not very accurate but they will leave grade school with a concept of topology (mathematical anyhow). hmmm, is probability a Topology? See where you can go with simple concepts The reason for calling any subset an "open subset" is historical. It indicates a generalization from the specific, from open subset to subset ("open subset"). Just as I would call cjem's definitions of neighborhoods, "neighborhoods" and a 4-dimensional space in physics (which is nonsense in the context of a generally accepted notion, and definition, of 3D space) a 4-dimensional "space." Personally, I believe a lot of confusion (and arrogance) arises when old terms are used with new meanings. Quotes is a way around this, which for practical reasons I don't advocate. OP definition is also a basis for discussion, as anyone can understand it. From Wolfram math World: Topology -- from Wolfram MathWorld "There is also a formal definition for a topology defined in terms of set operations. A set X along with a collection T of subsets of it is said to be a topology if the subsets in T obey the following properties: 1. The (trivial) subsets X and the empty set emptyset are in T. 2. Whenever sets A and B are in T, then so is A intersection B. 3. Whenever two or more sets are in T, then so is their union." My definition encompasses above. 1). 2), and 3) are a "specific subset." If you add "open subsets" = open subsets you have Euclidean geometry. And other definitions of "open subset" if you hit them cold can be quite baffling and leave you with no sense of perspective, and given the nature of mathematics and research (need a research topic, expand the definition of an existing concept, explore a few implications, and you are on your way), I'm sure there will be endless variations of "Topology." Please give an example of Topology not covered by my definition.
September 4th, 2017, 09:33 AM   #5
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 Please give an example of Topology not covered by my definition.
That's easy.

What happens if the set X is empty?

FYI the set X is usually called a topological space and the set T a topology on X.
X by itself is not a topology.
This brings out the fact that there are, in general, many topologies on any given set.

As Maschke said.

Attention to detail is everything in definitions.

I forget who said something like.

The only hard part of maths is getting the definitions right.
Once you have done that the rest is easy.

Last edited by studiot; September 4th, 2017 at 09:38 AM.

September 4th, 2017, 10:24 AM   #6
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 Originally Posted by studiot That's easy. What happens if the set X is empty? Trivial, covered FYI the set X is usually called a topological space and the set T a topology on X. Topology -- from Wolfram MathWorld A set X along with a collection T of subsets of it is said to be a topology https://en.wikipedia.org/wiki/Topology "---Formally, let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:---"

We are giving a simple, all encompassing, definition of "Topology." You can add whatever specifics you like.

September 4th, 2017, 04:05 PM   #7
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Quote:
 Originally Posted by studiot That's easy. What happens if the set X is empty? Trivial, covered
Where did you cover it?

 September 8th, 2017, 06:01 AM #8 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 896 Given a set, X, a "topology on X" or a "topology for X" is a collection of subsets of X such that 1) X is in the collection. 2) The empty set is in the collection. 3) If A and B are in the set them $\displaystyle A\cap B$ is in the collection. (It follows from this that the intersection of any finite number of sets in the collection is in the collection.) 4) The union of any sub-collection of sets in the collection is an the collection. (That does not have to be finite or even countable.) Some people will argue that (1) and (2) are not necessary because they follow from (4) using the "empty" sub-collection. That requires using the fact that "false implies A" is a true statement for any A which some other people prefer not to use. The sets in this collection are the "open sets". "Closed sets" are the complements of "open sets". It is also possible to define a topology by reversing the use of "finite" and "any" in (3) and (4). That gives a collection of "closed sets" and "open sets" are then the complements. Last edited by skipjack; September 10th, 2017 at 12:00 AM.
 September 8th, 2017, 11:32 AM #9 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 In my opinion, everything Country Boy writes is covered in OP, with room for generalization. If you wish (can) memorize Country Boy's defininition, by all means do so. or just look it up. In any event, I appreciate the Contribution. We will ignore the difficulties with "Empty Set."
 September 9th, 2017, 01:17 PM #10 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 896 Given any set, X, the "set and all of its subsets" is certainly a topology, called the "discrete topology". At the other end of the spectrum the "indiscrete topology" for any set, X, consists only of X and the empty set.

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