Looking for some feedback for solutions to select exercises from a basic Analysis course. All comments welcome!
- Determine whether or not each subset of $\mathbf{R}^2$ is compact. Briefly justify your answers.
- $A = \{(x, y)~ | ~ x^{2} + y^{2} \leq 4 \} - \{(x, y)~|~ x^{2} + y^{2} \leq 1 \}$
A is not compact. To be compact, a subset of $\mathbf{R}^n$ must be both bounded and closed. $A$ is bounded, but the point $(1,0)$ prevents $A^{c}$ from being open. Hence, $A$ is not closed, so $A$ is not compact.
- $B = \{(x, \sin x)~ |x \in \lbrack -2\pi, 2\pi \rbrack\} \cap \{(x, y)~|~ y\geq 0 \}$
B is compact. B is bounded above and below, and is therefore bounded. $B^{c}$ is also open, making B closed. Hence, B is compact.
- Let $X$ and $Y$ be compact subsets of a metric space. Prove that $X \cup Y$ is compact.
Let $U_{\infty}$ be an open cover for $A \cup B$. Then $U_{\infty}$ is an open cover for $A$ and for $B$. Since $A$ is compact, the union of some finite subset of $U_{\infty}$ $\{U_{1}, \ldots ,U_{k}\}$ covers $A$. Similarly, since $B$ is compact, the union of some finite subset of $U_{\infty}$ $\{U_{k+1}, \ldots, U_{n}\}$ covers $B$. Taking the union of both sets produces a finite sub cover $\{U_{1}, \ldots ,U_{k}, U_{k+1}, \ldots, U_{n}\}$ for both $A$ and $B$. Hence $A \cup B$ is compact.
You have to justify the assertions you're making. "$(1,0)$ prevents $A^c$ from being open" is not a rigorous argument. Why does this point do this? ("every neighborhood of $(1,0)$ ... ")
What are bounds for $B$? (saying "from above and below" is too informal, expecially since this is a subset of the plane, i.e. what about "left and right"?) Why is $B^c$ open?
The last one (with $X$ and $Y$) looks good.