I recently began learning about topology in terms of neighborhoods, open and closed sets, boundary and limit points. I'm trying to reconcile this with the view of topology as "rubber sheet geometry".
In particular, I'm picturing the set of points comprising the interior and boundary of a circle on the plane. If I understand correctly, this would be a closed set, and also an open set because the rest of the plane is closed.
From the rubber sheet view, I understand that the plane is pretty much like the surface of a sphere, and that if you add a "handle" to the sphere, you get a surface that's equivalent to a torus.
How do we express this "handle" concept in the language of point set topology?
Best Answer
The interior of the circle together with its boundary is a closed set. It's not also open, because the "rest of the plane" isn't closed - it's open.
The plane isn't quite like the surface of a sphere. It is near any point, but on the plane you can find an infinite sequence of points that never cluster anywhere. On the surface of the sphere that's impossible. (Of course this depends on a precise definition of "cluster").
To talk rigorously about the handle concept you have to do a fair amount of work. You define paths as continuous functions from the circle (just the boundary) into the space. Then you make precise what it means to "shrink a path to a single point". Then you have a (kind of) handle when you have a path that can't shrink to a point.