Is this generalization equivalent to usual Aposyndetic

[kmitza]

TL;DR

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.

 

[andrewkirk]

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.

 

[kmitza]

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.