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Problem 1. Please, answer the following questions. Please, do not forget that you must explain your...

Problem 1. Please, answer the following questions. Please, do not forget that you must explain your answers!! All sets below are subsets of Rd.

(a) Assume A and B are open sets, what can you say about set A + B, where

A + B = {a + b : a ∈ A, b ∈ B}.

(b) Assume A is an open set what can you say about set Pe1A, where Pe1 (A) is an orthogonal projection of A onto subspace orthogonal to the first basis vector e1, i.e.

Pe1 (A) = {(0, a2, . . . , ad) : (a1, a2, . . . , ad) ∈ A}.

(c) Is it true that a subset of a compact set is compact?

(d) Assume A is a compact set, what can you say about Pe1A?

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