use the following concepts: In the metric space Rn with the usual Euclidean metric, we can define a linear structure by setting (x 1 ,…., x n ) + ( y 1 ,…., y n ) = (x 1 + y 1 ,…., x n + y n ) And λ...

use the following concepts: In the metric space Rn with the usual Euclidean metric, we can define a linear structure by setting

(x₁,…., x_n
) + ( y₁,…., y_n
) = (x₁
+ y₁,…., x_n
+ y_n
)

And

λ (x₁,…., x_n
) = (λx₁,….,λx_n
)

for arbitrary points x = (x₁, …, x_n) and y = (y₁, …, y_n) in Rn and for λ ∈
.

If A, B ⊆
ⁿ
and λ ∈
, we define A + B = {a + b: a ∈ A and b ∈ B} and λB = {λb: b ∈ B}. If A consists of a single point, say p, then we often write p + B instead of A + B. The set p + B is called a translate of B. The set λB is called a scalar multiple of B. More generally, if λ ≠ 0, the set p + λB is said to be homothetic to B.

(a) Prove that each set homothetic to an open set is open.

(b) Prove that each set homothetic to a closed set is closed.

(c) Prove that A + B = ∪_a
∈
A
(a + B) = ∪_b
∈
B
(A + b).

(d) Prove or give a counterexample: If A is open, then for any set B, A + B is open.

(e) Prove or give a counterexample: If A and B are both closed, then A + B is closed.