For a set M ⊂ R^2, we define its shadow (projection) on the x-axis to be the set in R defined as

M_x={ x: there is some y so that (x,y) ∈ M }

a) Find a set M, so that M_x is open (in R) but M is not open (in R^2).

b) Suppose K is a compact set in R^2. Prove that K_x is also compact in R.

Hint: Either prove …first "if A is open in R, then A × R is open in R^2", then start with an open cover C of K_x and create a cover for K.

Or: start with a sequence (x_n) in K_x and use that K is sequentially compact to show that (x_n) has a cluster point in K_x.

c) Find a closed set N in R^2 such that N_x is not closed (in R) .

If anyone helps me, I will be thankful.