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Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply connected and has non-degenerate intersection form. Suppose we 'approximate' the given manifold by a sequence of compact subsets of $\mathbb{R}^{m}$ such that:

$$N_{i}\subset M, |\dim(N_{i})-4|<\frac{1}{i},\mu(M-N_{i})<\epsilon_{i},\epsilon_{i}\rightarrow 0$$ Here we are using Hausdauff dimension.

I asked an Geometric Analysis Professor on this problem, and he said the limit of $N_{i}$ might not tell us anything about $M$, especially $N_{i}$ has far worse analytical properties than $M$. So my question is - can we put a nice differentiable structure on $N_{i}$ such that the above limit makes some sense? For example, the unit real line may be `approximated' by various fat cantor sets. This question is quite unclear to me, so I wish to discuss at here.

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