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## Homework Statement

Let X=R^2 and the distance be the usual Euclidean distance. If C and D are non-empty sets of R^2 and we have:

C+D := {y ϵ R^2 | there exists c ϵ C and dϵD s.t c+d = y}

A) What is C+D if the open balls are C= ball((0.5,0.5);2) and D=ball((0.5,2.5);1)

B) Same as A) expect D is now a closed ball

C) same as a) except D={(l,-1)|l ϵ R}

D) Is the following true? If C,D are non-empty subsets of R^2 s.t C is open, then the sum C+D is open.

## Homework Equations

## The Attempt at a Solution

A) It to, C+D is just the union set of all points in C and D s.t. for all c ϵ C and dϵD, c + d = x.

I think this set is open, but not sure how to describe it in more detail. Any ideas?

B) Same description as A but with boundary too?

C) not sure

D) I think this is false since C could be open and D could be closed, and so their sum would not be open.