1) Let A, B be subsets of R^n that are pathwise connected with A interset B not empty. Prove A U B is pathwise connected.

2) Prove S = {(x,y) in R^2 : either x or y is rational number} is pathwise connected.

I have been studying for two weeks, and will continue to study tonight, but I simply cannot study some problems that I do not have the answers for.

Thank you!

2. #1 is easy. If x & y are in AUB then you have two cases.
If both x & y are in A or in B then there is an arc from x to y. WHY?
If x is in A and y is in B then there is a z in A intersect B. There is an arc from x to z and there is an arc from z to y. The union of those connects x to y with a path.

For #2, I would need to know what theorems about paths(arcs) you have.

3. So far, our theormes on pathwise connectedness are:

1. If A is pathwise connected, and f:A->R^n is continuous, then f(A) is pathwise connected.
2. If A is convex, then A is pathwise connected.
3. A is pathwise connected iff A is an interval.

I think that is all so far.