Question

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

(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 P_e1A, where P_e1 (A) is an orthogonal projection of A onto subspace orthogonal to the first basis vector e1, i.e.

P_e1 (A) = {(0, a₂, . . . , a_d) : (a₁, a₂, . . . , a_d) ∈ 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 P_e1A?

Answer 1