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- 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.

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.