Problem 1: Find the following partition of the set of letters

                A a B b C c D d E e F f G g H h I i J j K k L l M m N n O o P p Q q R r S s T t U u V v W w X x Y y Z z

into subsets in each of the following cases:

1. A subset contains a letter *  if an only if contains all the letters # such that there is  a continuous function from #  to *.
2. A subset contains a letter *  if an only if contains all the letters # such that there is  a homeomorphism from #  to *. (a homeomorphism is a continuous bijective function with continuous inverse)
3. All the letters * that can be isotoped into  #  in the plane R².
4. All the letters * that can be isotoped into  #  in space, that is, in R³.

A letter can be isotoped into other if there is a continuous deformation (like that of the doughnut into a coffee cup) such that in each "frame" determines a homeomorphism from the first letter to the figure in the frame.

You do not have to write formal proofs but a few words explaining your choices is necessary of a perfect score. (Since this problem has many parts, the grading  is 5 points for 1.1 and 1.2 5 points for 1.3 and 1.4)

Problem 2: Write down the definitions of : open set in Rⁿ , neighborhood of a point  and limit point.

Problem 3: Prove that the set of all points (x,y,z) in R³  such that x.y.z ≠ 0 is open.

Problem 4: Prove that for each x in  R² and r>0, the ball of center x and radius r is open. (Yes, we did it in class...)