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Would someone check my solution to this exercise:

Exercise. Determine whether the following subsets of $\mathbb{R^2}$ are open, closed, and/or bounded.

  1. $A=\{\|x\|\le1\}$
  2. $B=\{\|x\|=1\}$
  3. $C=\{\|x\|\lt1\}$
  4. $D=\{\text{the x-axis}\}$
  5. $E=\mathbb{R^2}-\{\text{the x-axis}\}$
  6. $F=\{(x,y):x \text{ and } y \text{ are integers}\}$
  7. $G=\{(1,0),(1/2,0),(1/3,0),\dots\}$
  8. $H=\mathbb{R^2}$
  9. $I=\emptyset$

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 $x$.

  1. 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 $A\subseteq B(0,2)$
  2. Closed (but not open) and bounded (as above).
  3. Open but not closed (because points $x$ with $\|x\|=1$ intersect $C$). $C$ is bounded (a ball of radius $2$ contains $C$)
  4. Closed, not open, not bounded.
  5. Open, not closed, not bounded.
  6. Closed, not open, not bounded.
  7. Closed, not open, bounded ($\subseteq B(0,2)$)
  8. Open, closed (vacuously true), not bounded.
  9. Open (vacuously true), closed, bounded.
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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.

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  • $\begingroup$ Thank you! A ball around 0 must include positives in the set so it's not closed. CorrecT? $\endgroup$ – saadtaame Jul 19 '13 at 12:51
  • $\begingroup$ That's correct! $\endgroup$ – Cameron Buie Jul 19 '13 at 12:52
  • $\begingroup$ Are limit points the same as the ones we study in calculus ($lim$) $\endgroup$ – saadtaame Jul 19 '13 at 12:54
  • $\begingroup$ Indeed, the concepts are closely related. We can talk about limits of functions as we approach limit points of their domain, for example. That question is a bit too broad to be answered fully in a single comment, but I suspect that it has been asked before on this site, so you might take a look and see if you can find it. $\endgroup$ – Cameron Buie Jul 19 '13 at 12:56
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For $7.$ the sequence $(1/n,0)$ is convergent to $(0,0)\not \in$ the set.

For $1.2.3)$ you can use also the continuous function $f\colon\mathbb R^n\rightarrow \mathbb R,x\mapsto ||x||$ and the fact that the preimage of closed (open) set by a continuous function is closed (open).

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  • $\begingroup$ Thanks! It's the first exercise in the book. So no resort to continuity or advanced stuff. $\endgroup$ – saadtaame Jul 19 '13 at 12:48

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