Open Sets

Distance -> Open ball -> interior points, boundary points, exterior points -> open sets -> closed sets…… this is a line in real analysis view… while in topological definition open set is the most basic conception, and the interior of set A is defined with open set: the interior of set A is the largest open set contained in A…… maybe from the above summary, it seems can state topology is more abstract than real analysis.

Structure

Heard of that something has a structure, like the mechanical system, so the people look for the so called structure; then often heard of another way that something is a kind of structure, like point sets, so the people look for the properties of the structure; while recently realized that structure is an angle to view something, like the one, two or three dimensional geometry space, we look at it based on the points, lines and the relations between them, which is called geometry structure, while we look at the higher dimensional real space from the view of sets, which is a kind of generalized points or line, it’s called topology structure, and we can do that from other views or to say structures, but they may be not so clear as the former set view.

Rectangles and cubes

Rectangles and cubes are the building blocks of real analysis, whose objective is n dimensional real space, which we are not familiar with. We always study something new from the old which can be treated as the first step, just like we use the two or three dimensional space concepts areas and volumes as the preliminaries forward to higher dimension. Any basic geometry methods should be able to be used here, but may be not so useful or so clear. Then the more general description, measure, is developed.

"Disjoint" and "Almost Disjoint" in Math (Real Analysis)

Two sets are "disjoint" means the distance between them are larger than 0, while when we say two sets are "almost disjoint", it seems to refer rectangles but not the general sets, and it means the interiors of the rectangles are disjoint.