Here’s a fun question that was floating around Mathcamp last week: find a finite topological space which has a nontrivial fundamental group. One answer to this question after the jump.
One example is a space S with 4 points, two of which are open and two of which are closed. First, consider the line with the origin doubled. Now quotient out by setting all positive points equal to each other, and all negative points equal to each other. This gives a four point space S.
There’s a map from the circle to S given by sending your favorite two points on the circle to the closed points, and the two open intervals between them to the open points. It is not difficult to see that this cannot be extended to the disc. A better proof is to exhibit S’ the universal cover of S. The space S’ looks like:
The points in the middle column are closed. The points in the other two columns are open, and the closure of any such point contains the two nearest points in the middle column. S’ is not contractable, but any compact (i.e. finite) subset of it is contractable, so it is simply connected. Hence since the deck transformations of S’ just come from shifting up and down.
Here are two more fun problems: find all the homology and homotopy groups of this 4 point space.