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 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 n-space: A set S in n-space is open if every point in S has a neighborhood in n-space consisting only of points in S. 1a) Topology (relative) on a set S in n-space: 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
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Quote:
 Originally Posted by zylo Topology on a set S is the definition of an open set in S.
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.

Quote:
 Examples 1) Topology on n-space: A set S in n-space is open if every point in S has a neighborhood in n-space consisting only of points in S. 1a) Topology (relative) on a set S in n-space: A set U in S is open if every point of U has a neighborhood in S consisting only of points in U.
Yes, with "neighborhood" defined in terms of the distance function in n-space.

Quote:
 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.
No, this is NOT a topology because the entire set of all marbles in the box does not have this property.

Quote:
 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.
Same objection as before.

Quote:
 Associated with (dependent on) definition of open are closed, complement, boundary, etc.
Do you have a question? Are you asking if these are true?

September 8th, 2015, 06:45 AM   #3
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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.
1) is a fundamental error of set theory. A set cannot be a member of itself- it is a circular definition.
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
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Quote:
 Originally Posted by zylo 1) is a fundamental error of set theory. A set cannot be a member of itself- it is a circular definition.
No. Country Boy isn't claiming that S is a member of itself, but that S is a member of the topology.

Also, there are set theories which allow sets to be members of themselves, like the Quine atom Q = {Q}. ZF doesn't, though.

Quote:
 Originally Posted by zylo 2), 3), and 4) are sophistries of set theory irrelevant to fundamental definition of Topology.
If you don't have the union and finite intersection properties, you don't have a topology at all but a very different creature! These are major foundational properties, without which most of topology fails.

Charitably, you may have misunderstood the points raised above.

September 8th, 2015, 11:31 AM   #5
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Quote:
 Originally Posted by CRGreathouse No. Country Boy isn't claiming that S is a member of itself, but that S is a member of the topology.

Quote:
 Originally Posted by CRGreathouse Also, there are set theories which allow sets to be members of themselves, like the Quine atom Q = {Q}. ZF doesn't, though.
.
The topic is not axiomatc set theory, which doesn't even derive from a definition of topology.

Quote:
 Originally Posted by CRGreathouse If you don't have the union and finite intersection properties, you don't have a topology at all but a very different creature! These are major foundational properties, without which most of topology fails.
union, intersection, open, closed, complement, boundary etc are foundational properties from set theory, not derivable from the definition of topology. Of course topology applies to sets, but you need a definition of topology to start.

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 re-read 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. Thanks from zylo
September 9th, 2015, 06:43 AM   #7
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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.
Is it a definition of topology, or an example of something (unspecified) called topology? The OP appears to satisfy this definition, but not transparent.

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
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Quote:
 Originally Posted by CRGreathouse 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.
I said topology was the definition of an open set, however you wish to define open set and all that entails. To illustrate, I gave some examples. That is simple, accurate, easily remembered, and adds structure.

Unfortunately, the discussion got bushwacked by someone introducing topological spaces without saying so.

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