In an excercise of Spivak - Calculus on manifolds it's asked to construct a subset $A$ of $[0,1]×[0,1]$ such that it contains at most one point of each horizontal and vertical line and whose boundary is $[0,1]×[0,1]$.
It turns out that we can construct such a set as follows:
Divide the closed square $[0,1]×[0,1]$ into four open squares $(0,1/2)×(0,1/2)$, $(1/2,1)×(0,1/2)$, $(0,1/2)×(1/2,1)$ and $(1/2,1)×(1/2,1)$ and pick one point from each of those squares in such a manner that it's not at the center of the square and the horizontal and vertical lines passing through that point don't pass through any of the other three points. Now divide again each of the open squares into four open squares and repeat the procedure. Continuing in this fashion I think we achieve our goal (I followed a hint ok the book).
This excercise made me realize that this set $A$ is one of a class of sets I've never thought of, namely the class of sets that are proper subsets of their boundaries, so I wonder what are sufficient conditions for a set to be of this class.
I've noticed that it's necessary for a set $A$ to be of this class that $A$ is neither open nor closed since if it were open all of its points should be interior and that can't happen because all of its point are points of its boundary. On the other hand it can't be closed because in that case it would contain its boundary and hence it wouldn't be a proper subset of its boundary. This is all I've got but obviously these conditions are not sufficient.
I'm looking for a characterization of this kind of sets (preferably in an arbitrary topological space), some other examples are welcome since they might help to abstract their common characteristics.
Thanks