I want to construct a nowhere dense set of positive measure by using the Smith-Volterra-Cantor method but subtracting closed sets intead of open sets.
Suppose we start with the open interval (0,1). Let $\alpha=1/2$. Now remove $U_1=[1/2-\alpha/4,1/2+\alpha/4]$. The measure $\mu(U_1)=\alpha/2$. For each of the two remaining open sets, subtract a closed from the center of width $\alpha/8$, thus a total measure of $\alpha/4$. Repeating this process results in removing a total measure of $\alpha/2+\alpha/4+\dots=\alpha$.
Does this give me a set of positive measure just as with subtracting closed intervals? I suspect not, since I have never heard of this, but what's the exact function of removing a closed set instead of an empty set?
Or, if I can do this, can I now consider this to be an open set that I can choose an admissible partition of unity for?