Question

1. Open and closed sets a) Are the following sets open or closed or none of the two? Justify your answer. (4 points) $A_1 = [-1, \infty) \times (-\infty, 1]$, $A_2 = A_1 \setminus \{(0, 0)\}$, $A_3 = \mathbb{N}$, $A_4 = \mathbb{R} \setminus \mathbb{N}$. b) (4 points) (Consider two open sets A and B. Which of the following statements are correct? Give a proof or a counterexample: $A \cap B$ is open, $A \cap \bar{B}$ is open ($\bar{B} = B \cup \partial B$ is the closure of B.

          1. Open and closed sets
a) Are the following sets open or closed or none of the two? Justify your answer. (4 points)
$A_1 = [-1, \infty) \times (-\infty, 1]$, $A_2 = A_1 \setminus \{(0, 0)\}$, $A_3 = \mathbb{N}$, $A_4 = \mathbb{R} \setminus \mathbb{N}$.
b) (4 points) (Consider two open sets A and B. Which of the following statements are correct?
Give a proof or a counterexample: $A \cap B$ is open, $A \cap \bar{B}$ is open ($\bar{B} = B \cup \partial B$ is the
closure of B.

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1. Open and closed sets
a) Are the following sets open or closed or none of the two? Justify your answer. (4 points)
A1 = [-1, ∞) × (-∞, 1], A2 = A1 ∖{(0, 0)}, A3 = ℕ, A4 = ℝ∖ℕ.
b) (4 points) (Consider two open sets A and B. Which of the following statements are correct?
Give a proof or a counterexample: A ∩ B is open, A ∩B̅ is open (B̅ = B ∪∂ B is the
closure of B.

Added by Frank C.

University Physics with Modern Physics

Hugh D. Young 14th Edition

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Solved by Expert Supreeta N

Step 1

Any sequence that converges to a point greater than or equal to 1 will be contained in A1. - A2=(-∞,1] is closed because it contains all its limit points. Any sequence that converges to a point less than or equal to 1 will be contained in A2. - A3 = N (the set of Show more…

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a) Are the following sets open or closed or none of the two? Justify your answer. (4 points) A1=[1,âˆž), A2=(-âˆž,1], A3 = N, A4 = R. b) (4 points) Consider two open sets A and B. Which of the following statements are correct? Give a proof or a counterexample: A âˆ© B is open, A âˆª B is open, B = B', B is the closure of B.

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