I am trying to better understand mutually dense sets in a topology. I have found very little good information on this and most of that has dealt with ordered sets, such as the mutual density of the rationals and the non-rationals. However I have also seen it stated that C and D are mutually dense sets if they are contained in each other's closures, so that $C \subseteq \overline{D}$ and $D \subseteq \overline{C}$, which is seemingly more general and of more interest to me than definitions based on order. I am particularly interested in the relations of limit points of such sets. Can anyone give me some good explanations, links, salient facts, references or other enlightenment?
Thanks!
In response to the first answer below by Adam: Suppose we have C and D as disjoint sets but each including the other’s boundary. (Thought experiment: Imagine two congruent, two-dimensional sets on two sheets of paper, one above the other so they are disjoint, but infinitely close in the third dimension so that each contains limit points of the other. Then squash the third dimension down so that the points intermingle, but retain their set memberships.) Then it seems to me that $C \cap D = \emptyset$ and since $d(C) \cap D = D$ we have $D = \overline{C}\setminus C$ and hence $D \subset \overline{C}$. By symmetry then also $C \subset \overline{D}$.
@Adam - "congruent" was just being sloppy, meaning they fit on top of one another on the two sheets. I was just trying to give a helpful visualization of the idea, not a mathematical argument.
@Joriki - Thanks for the link. Yes, I'm afraid it is broad. I'm trying to learn more about the necessary and sufficient conditions for sets to be mutually dense, and about the ramifications of their being so. I am having trouble finding much on it, and in particular much that treats with things more generally than a quick example on the real line. For example, it is not obvious to me why sets being subsets of the others' closure would entail anything like a necessity for their points to be such that between any two points of one there was a point of the other and vice versa in more than one dimension.