[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, n points from the plane.  You tell me the homology groups of the resulting space.

Read the rest of this entry »

Advertisements

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. 

Read the rest of this entry »