Dimension has no meaning when you talk about sets.

Venn diagrams are a nice way to demonstrate some basic principals. Each point on the diagram is supposed to represent an element, so a set looks like a bunch of these (circle is usually used).

I have no idea what you mean with the X, Y and Z.

A neighborhood is a set which has a given point in its interior.

If you want to understand topology properly I suggest you first look at metric spaces. Topology is to metric spaces, what metric spaces are to the Euclidean setting.