February 26, 2011
[This homology adventure will, unfortunately, have no pictures. I want to make some, but my bamboo pad is being irritable.]
Here’s the game: Take the plane. I remove, say, points from the plane. You tell me the homology groups of the resulting space.
February 25, 2011
For the seasoned mathematician, this is a relatively obvious statement. In a complete metric space, the sequence will even have a limit, and thus trivially be bounded by just taking a ball around the limit and noting only finitely many things lie outside. But when a student asked about this question and the question did not specify the space was complete, I thought that I might be able to construct a counterexample — you know, the points are close together but just far enough apart such that they go off to infinity.