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Flashcards in S3 Deck (10)
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1

Let A ⊂ ℝ be a non-empty subset. 

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

False.

Take, for example, A = [0, 1[ ∪ ]1, 2].

The statement would be true for an interval. 

2

Let A ⊂ ℝ be a non-empty subset. 

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

False

Take A =]0, 1] ∪ [2, 3[.

3

Let A ⊂ ℝ be a non-empty subset. 

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

False.

Take A =]0, 1[, so that ∂A = {0, 1}. 

4

Let A ⊂ ℝ be a non-empty subset. 

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

False.

√4 = 2 ∈ ℚ  is the supremum of A. 

 

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. 

True.

The boundary is ∂A = ¬A ∩ ¬(ℝ\A).

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

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. 

False.

Take A = {1}. We have 1 ∉ Å = ∅. 

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. 

False.

Take A={1}.

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

8

Consider the set E.

E ∪ {1} is closed. 

True.

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

9

Consider the set E.

E has an infinite number of isolated points. 

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. 

10

Consider the set E.

E° = E. 

False.

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