For any CW complex, the boundary of the boundary is the empty set. Outside of CW complexes, are there sets for which the boundary of the boundary is nonempty? Examples in one or two dimensions preferred.
EDIT: Based on the comments below I'm trying to make the question more precise:
- The notion of boundary requires a subspace S < X. It is defined by $\partial S = \overline{S} \cap \overline{X \backslash S}$ (taken from Munkres or Wikipedia)
- X should be Hausdorff and have countable basis. The Cantor set, being homeomorphic to {0,1}N
is not countable basis.
It could be Hausdorff + countable basis + subspace of Rn are enough to ensure ∂2=0. Originally, I was hoping to find spaces "close" to being CW complexes for which ∂2 ≠ 0.
EDIT: The square of the boundary map should be defined by $\partial^2 S = \overline{\partial S} \cap \overline{S \backslash \partial S}$ - this is the boundary of $\partial S$ relative to S. I am looking for topological subspaces $S \subset X = \mathbb{R}^n$ for which $\partial^2 S \not = \varnothing$. Maybe Hausdorff and countable basis is enough to ensure this can't happen. Certainly this rules out the disk in R2. Perhaps such a counterexample could arise as "limit" of CW complexes (e.g. in the Hausdorff metric).
EDIT:There is still a defect in the above definitions. What definition do you suggest so that the boundary of the boundary of the disk is empty? Can this definition be extended to arbitrary sets? Will the boundary of the boundary still be empty for all sets?
