Is this generalization equivalent to usual Aposyndetic

In summary, the speakers discuss the definitions of "connected, metric space" and "zero set aposyndetic." They question whether the two are equivalent and provide a potential counterexample using dense, connected subsets. The speaker clarifies that the definition of continuum should also include compactness, and that zero set aposyndetic can only work with non-intersecting sets.
  • #1
kmitza
17
4
TL;DR Summary
I want to check if this "new" notion is equivalent to a well known one and I feel like I am missing an obvious counterexample
For some basic definitions we call connected, metric space a continuum and we say that continuum is aposyndetic if for every pair of points p,q exists a subcontinuum W such that $p \in int(W) \subset W \subset X \setminus \{q\}$ similarly I introduce a notion of "zero set aposyndetic" as:
X is aposyndetic if for every two empty interior connected subsets U,V exist W such that $U \subset int(W) \subset W \subset X\setminusV$
I want to check if the two are equivalent as it is obvious that "zero set aposyndetic" implies aposyndetic but I feel intuitively that the other direction might not be true, however I can't see the counterexample nor the proof.
 
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  • #2
If we could find two dense, connected subsets, that should serve as a counterexample, since the only open set containing either of them will be the whole space X, so it must overlap the other subset.

Taking ##\mathbb A## to represent the algebraic numbers, I think if we take ##X=\mathbb R^2, U = X - \mathbb A^2, V = \mathbb A^2 - \mathbb Q^2## it might work.

Certainly U and V are dense in X. I think they may also both be connected (not path-connected, but that doesn't matter). This doesn't work with ##X=\mathbb R## but I think it works in two dimensions.
 
  • #3
Hi, first of all thank you for your answer :)
Secondly I now notice I made a wrong definition of continuum... It should also be compact.
Anyways your approach works if I work a dense set and any other set to go along with it.
Also just for clarification zero set aposyndetic can only work with non intersecting sets I didn't state that in the definition as I made it up so I was being careless.
 

1. Is the generalization equivalent to usual Aposyndetic?

This question is asking if the generalization follows the same format or structure as the typical Aposyndetic. The answer would depend on the specific generalization and how it compares to the usual Aposyndetic.

2. What is the usual Aposyndetic?

The usual Aposyndetic refers to a specific format or structure commonly used in scientific or mathematical writing. It involves listing items or ideas without any conjunctions connecting them.

3. How can I determine if a generalization is equivalent to usual Aposyndetic?

To determine if a generalization is equivalent to usual Aposyndetic, you can compare the structure and format of the generalization to the typical Aposyndetic. If they follow the same format, then the generalization can be considered equivalent.

4. Are there any benefits to using Aposyndetic in generalizations?

Some potential benefits of using Aposyndetic in generalizations include creating a concise and clear statement, emphasizing the individual items or ideas, and allowing for easy comparison between them.

5. Can Aposyndetic be used in other types of writing besides generalizations?

Yes, Aposyndetic can be used in various types of writing, including scientific and mathematical writing, as well as creative writing. However, it is important to use it appropriately and not overuse it in a way that may affect the clarity or flow of the writing.

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