I have been recently studying graph theory. Graphs connect points on a 2d surface with lines. Is there a name and theory for structures which connect points in 3 dimensional spaces with surfaces? Also are analogues in higher dimensions studied, such as structures where (n-1) dimensional objects connect points in n dimensional spaces?
Further I have been thinking about structures where (n-1) dimensional objects connect (n-2) dimensional objects in n dimensional space. eg. Planes which connect lines in 3 dimensional space . Have such structures been explored? Thank you.
Yes. These higher-dimensional analogs of the building blocks of graph-theory are generally called "simplicial complexes," and are at the heart of much of algebraic topology.
The usual structure (in topology) is a bit more inductive than what you describe: when you connect together 3 points, you also have to connect together their edges pairwise, to get 3 vertices, the 3 edges, and the one face of a triangle. And there are variations which allow, as in graph theory, for there to be multiple "edges" between pairs of points, multiple "faces" sharing corresponding points (or edges), etc. What structure is most interesting for study usually depends on what kind of phenomenon you're trying to model.