Would someone check my solution to this exercise:
Exercise. Determine whether the following subsets of $\mathbb{R^2}$ are open, closed, and/or bounded.
- $A=\{\|x\|\le1\}$
- $B=\{\|x\|=1\}$
- $C=\{\|x\|\lt1\}$
- $D=\{\text{the x-axis}\}$
- $E=\mathbb{R^2}-\{\text{the x-axis}\}$
- $F=\{(x,y):x \text{ and } y \text{ are integers}\}$
- $G=\{(1,0),(1/2,0),(1/3,0),\dots\}$
- $H=\mathbb{R^2}$
- $I=\emptyset$
Solution.
First, define a set $A$ to be open if every $x\in A$ is an interior point and to be closed if every $x\notin A$ is an exterior point. LET $B(x,\epsilon)$ denote the open ball of radius $\epsilon$ centered at $x$.
- Let $x\notin A$. Then $B(x,\epsilon)$ is a neighborhood of $x$ with $\epsilon=\|x\|-1$ So $A$ is closed. $A$ is not open because points $x$ with $\|x\|=1$ have neighborhoods that contain points not in $A$. $A$ is bounded because $A\subseteq B(0,2)$
- Closed (but not open) and bounded (as above).
- Open but not closed (because points $x$ with $\|x\|=1$ intersect $C$). $C$ is bounded (a ball of radius $2$ contains $C$)
- Closed, not open, not bounded.
- Open, not closed, not bounded.
- Closed, not open, not bounded.
- Closed, not open, bounded ($\subseteq B(0,2)$)
- Open, closed (vacuously true), not bounded.
- Open (vacuously true), closed, bounded.
Best Answer
Except for number 7, you're correct. Is the origin an exterior point of $G$?
Incidentally, the only subsets of $\Bbb R^2$ (in the usual metric-induced topology) that are both closed and open are $\Bbb R^2$ and $\emptyset$. This is a fact you might want to keep in mind for future problems.