S3

Question 1

Let A ⊂ ℝ be a non-empty subset.

If sup(A) ∈ A and inf(A) ∈ A, then A is closed.

Answer 1

False.

Take, for example, A = [0, 1[∪]1, 2].

The statement would be true for an interval.

Question 2

Let A ⊂ ℝ be a non-empty subset.

If sup(A) ∉ A and inf(A) ∉ A, then A is open.

Answer 2

False.

Take A =]0, 1] ∪ [2, 3[.

Question 3

Let A ⊂ ℝ be a non-empty subset.

If A is open, then its boundary ∂A is empty.

Answer 3

False.

Take A =]0, 1[, so that ∂A = {0, 1}.

Question 4

Let A ⊂ ℝ be a non-empty subset.

If A = {x: 0 ≤ x² < 4, x ∈ ℚ}, then A has no supremum in ℚ.

Answer 4

False.

√4 = 2 ∈ ℚ is the supremum of A.

Question 5

Let A ⊂ ℝ be a non-empty subset. We denote by Å its interior, ¬A its closure and ∂A its boundary.

The boundary ∂A of A is closed.

Answer 5

True.

The boundary is ∂A = ¬A ∩ ¬(ℝ\A).

The closure of a set is closed and the intersection of two closed sets is closed.

Question 6

Let A ⊂ ℝ be a non-empty subset. We denote by Å its interior, ¬A its closure and ∂A its boundary.

If a ∈ A, then a ∈ Å.

Answer 6

False.

Take A = {1}. We have 1 ∉ Å = ∅.

Question 7

Let A ⊂ ℝ be a non-empty subset. We denote by Å its interior, ¬A its closure and ∂A its boundary.

If a ∈ ∂A, then a is a limit point of A.

Answer 7

False.

Take A={1}.

We have ∂A = {1} but 1 is not a limit point of A.

Question 8

Consider the set E.

E ∪ {1} is closed.

Answer 8

True.

E ∪ {1} is the closure of E and the closure of a set is closed.

Question 9

Consider the set E.

E has an infinite number of isolated points.

Answer 9

True.

E has infinitely many elements. They are all isolated since for each of them, we can

find a closed interval in R which contains only that point of E.

Question 10

Consider the set E.

E° = E.

Answer 10

False.

Since E contains only isolated points, E° = ∅.