Dense sets and the empty set

HallsofIvy

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I refer to the following defintions from the previously referenced Borowski and Borwein (1991 p 591) for "topology"

1. Point set topology: the branch of mathematics that is concerned with the generalization of the concepts of limits, continuity, etc to sets other than the real or complex numbers.

2. Algebraic topology: a branch of geometry describing the properties of a figure that are unaffected by continuous distortion such as stretching or knotting.

3. A family of subsets of a given set that constitute a topological space. The discrete topology consists of the entire power set, while the indiscrete topology contains only the empty set and the entire space.

I was only arguing that if T is the point set of a surface (ie the surface of a torus), T is dense in some other set. That is not to say that a topological space must contain a dense set.
You are again referring to a "dense set" as if "dense" were a property of a single set. That's not true. Set A is "dense" in set B if and only B is equal to the closure of A.

I don't see WHY you arguing "that if T is the point set of a surface (ie the surface of a torus), T is dense in some other set." It's clearly not true. The surface of a torus, or any manifold with surface, is a closed set. B= closure of A cannot be true in that case unless B= A itself.


Because of the reference to "all subsets of A", it is clear that "all sets contain the empty set" should have been "all sets have the empty set as a subset". However, since it is the case that the poster's statement "If A is a dense set, then all subsets of A are dense" is false the whole question becomes moot.
What were you referring morphism to?
 
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You are again referring to a "dense set" as if "dense" were a property of a single set. That's not true. Set A is "dense" in set B if and only B is equal to the closure of A.

I don't see WHY you arguing "that if T is the point set of a surface (ie the surface of a torus), T is dense in some other set." It's clearly not true. The surface of a torus, or any manifold with surface, is a closed set. B= closure of A cannot be true in that case unless B= A itself.



What were you referring morphism to?
I was referring to post #20, second paragraph where you state, "No set can be dense without saying in what set." Now you seem to be saying that T (as a closed surface) can be dense in itself. If so, this was my was my original position, although I was not specific in my first post regarding a closed surface. The question remains, does T as a closed surface also contain the empty set as a subset or must T be dense in another set which also contains the empty set?
 

HallsofIvy

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I was referring to post #20, second paragraph where you state, "No set can be dense without saying in what set." Now you seem to be saying that T (as a closed surface) can be dense in itself. If so, this was my was my original position, although I was not specific in my first post regarding a closed surface. The question remains, does T as a closed surface also contain the empty set as a subset or must T be dense in another set which also contains the empty set?
This is getting stranger and stranger! Every set is dense in itself. And yes, "No set can be dense without saying in which set." What I said about T was that, since T is closed itself, it cannot be dense in any larger set. Yes, it is true that T is dense in itself- which is precisely saying "in what set".

Finally, every set, whether "proper" or not, whether a topological space, ordered set or whatever, has the empty set as a subset. I don't see what that has to do with everything.
 
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This is getting stranger and stranger! Every set is dense in itself. And yes, "No set can be dense without saying in which set." What I said about T was that, since T is closed itself, it cannot be dense in any larger set. Yes, it is true that T is dense in itself- which is precisely saying "in what set".

Finally, every set, whether "proper" or not, whether a topological space, ordered set or whatever, has the empty set as a subset. I don't see what that has to do with everything.
Simply that if every subset of T (as 'morphism' says in post 24) is dense in itself, then the empty set must be dense in itself as a subset of T. Since the empty set contains no points of the point set T, this would appear to me to be a contradiction.
 
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quadraphonics

Simply that if every subset of T (as 'morphism' says in post 24) is dense in itself, then the empty set must be dense in itself as a subset of T. Since the empty set contains no points of the point set T, this would appear to me to be a contradiction.
No, the usual definition of a dense set is: "If set X is dense in set Y, any point in set Y can be 'well-approximated' by a point in set X". If set Y is the empty set, that is trivially true, since there are no points in Y to worry about.
 

HallsofIvy

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Simply that if every subset of T (as 'morphism' says in post 24) is dense in itself, then the empty set must be dense in itself as a subset of T. Since the empty set contains no points of the point set T, this would appear to me to be a contradiction.
No, the usual definition of a dense set is: "If set X is dense in set Y, any point in set Y can be 'well-approximated' by a point in set X". If set Y is the empty set, that is trivially true, since there are no points in Y to worry about.
Or, a little more generally, a set X is dense in set Y if and only if Y is contained in the closure of Y. As has been said repeatedly, every set, no matter what it is a subset of, is dense in itself because every set is contained in its own closure. The empty set is itself a closed set: The closure of the empty set is itself which certainly contains itself. That has nothing at all to do with whether it contains any points.
 
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Or, a little more generally, a set X is dense in set Y if and only if Y is contained in the closure of Y. As has been said repeatedly, every set, no matter what it is a subset of, is dense in itself because every set is contained in its own closure. The empty set is itself a closed set: The closure of the empty set is itself which certainly contains itself. That has nothing at all to do with whether it contains any points.
Thank you Quadaphonics and HallsofIvy. The key, at least to me, is that the empty set is in fact considered 'dense in itself.'
 
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HallsofIvy

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Since the definition of "closure" of a set is the set itself union all limit points, every set is contained in its own closure: every set is dense in itself. I don't see how that is "key" to anything!

The whole point of "density", typically, is to have some comparatively small set dense in a large set (as the countable rationals in the uncountable reals).
 

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