Provide an appropriate response.
-Which of the following space regions is (are) closed?
i. The hemispherical region centered at the origin with $z 0$ and radius $\mathrm { r }$ bounded by $0 \leq \mathrm { r } \leq \mathrm { r } _ { \mathrm { O } }$
ii. The $x y$-plane
iii. The half-space $x 0$
iv. Space itself
A) ii only
B) ii, iii, and iv
C) ii and iv
D) iv only

Question

Provide an appropriate response.
-Which of the following space regions is (are) closed?
i. The hemispherical region centered at the origin with $z > 0$ and radius $\mathrm { r }$ bounded by $0 \leq \mathrm { r } \leq \mathrm { r } _ { \mathrm { O } }$
ii. The $x y$-plane
iii. The half-space $x > 0$
iv. Space itself
A) ii only
B) ii, iii, and iv
C) ii and iv
D) iv only

Quizplus · Accepted Answer

- Region ii is closed because it includes its boundary (the xy-plane).
- Region iii is not closed as it does not include its boundary (the vertical plane x=0).
- Region iv is closed because it includes all its points and the boundary (which is the entire space).
- Region i is not closed because it includes its boundary (the hemisphere surface) but not its points on the boundary where z = 0.

Provide an Appropriate Response z>0z > 0z>0 And Radius r\mathrm { r }r

Provide an Appropriate Response $z > 0$ And Radius $\mathrm { r }$