It's not nearly that bad. Manifolds are more advanced than general topology. Here's the basic overview ... http://en.wikipedia.org/wiki/General_topology But the concept of boundary is very simple. You have a set of points in the plane, let's say the closed unit disk defined by x^2 + y^2 <= 1. Now for each point in the disk, imagine drawing a little circle around it. If a point has the property that x^2 + y^2 < 1, in other words the point is strictly inside the region bounded by the unit circle, you could find a tiny little circle around that point such that the entire circle is inside the region. We call that part the interior of the closed unit disk. Now if a point happens to be exactly on the circle; that is, it satisfies x^2 + y^2 = 1, then any circle you draw around the point must necessarily contain points both inside the region and points outside it. Any point of the region that has this property is called a boundary point. You can play the same game with the outside region x^2 + y^2 > 1. Points exactly on the unit circle are boundary points; and points "outside" the circle are technically interior points of the outer region. That's really everything there is to know about this ... the definition of the boundary of a set of points in the plane. If you have a set of points in the plane S, a point b in the plane is called a boundary point of S if any circle around b must contain both points of S and points of the complement of S. The set of all boundary points of S is called the boundary of S. Note that the definition of boundary point does not require the boundary point to be a point of S. A boundary point may be an element of S; or it may be an element of the complement of S. Either way, as long as any circle around the point must contain points of S and points of the complement of S; then we call that point a boundary point of S. It's a very clever definition, because it solves the circle problem! Everything you need to know is in what I wrote, so feel free to ask questions. Definitely no need to deal with manifolds or anything else.