Which of the following of subspaces of $\mathbb{R^2}$ are compact?

Question

Which of the following of subspaces of $\mathbb{R^2}$ are compact ?

(a) $\{(x, y) \in \mathbb{R}^2 : xy = 0$}.

(b) $\{( \frac{1}{n}, \frac{1}{n} )\in \mathbb{ R}^2 : n = 1, 2, . . . \} ∪ \{(0, 0)\}$.

(c) $\{(r_n \cos n\phi, r_n \sin n\phi) ∈ \mathbb{R}^2\}$, where $\phi \in [0, 2\pi)$ and $\lim r_n = +\infty.$

(d) $\{(x, y) \in \mathbb{R}^2 : x^ 6 + y^ 6 = 1\}$

my work:

(a) is closed but I am not sure about bounded

(b) is close the set of limit point $(0,0)$ in that set don't idea about bounded

(c) No idea

(d) this set is closed and bounded so compact

Answer 1

(a) is closed but i am not sure about bounded

• xy=0 correspond to x and y axis

(b) is close the set of limit point $(0,0)$ in that set don't about bounded

• it is bounded since $0<\frac1n\le1$

(c) No idea

• it seems unbonded

(d) this set is closed and bounded so compact

• yes I agree

Answer 2

(a) is not bounded (it's the axes).

(b) It's bound; it's a convergent sequence together with its limit.

(c) It's not bounded; the norms go to $+\infty$.

(d) Right.