Exercise 2.12

Exercise 2.12: Let $d$ be the euclidean metric on $\mathbb R^{2}$. Find every clopen set in the Euclidean topology induced by that metric on $\mathbb R^{2}$.

Incomplete Solutions

Olomana

Clopen sets occur in pairs, because the complement of a clopen set is also clopen. $X$ and $\emptyset$ are one such pair.

Assume there is another pair, two nonempty clopen sets $U$ and $V$ such that $V$ is the complement of $U$. Let $u_1$ be a point in $U$ and $v_1$be a point in $V$. Construct the midpoint $w_1$ by adding the coordinates of $u_1$ and $v_1$ and dividing by $2$. Since $U$ and $V$ are complements, $w_1$ must be in $U$ or $V$. If $w_1$ is in $U$, set:

$u_{2}=w_{1},v_{2}=v_{1}$

If $w_1$ is in $V$, set:

$u_{2}=u_{1},v2=w_{1}$

Lather, rinse, repeat. Construct a sequence of smaller and smaller segments $\{[u_{i},v_{i}]\}$. Note that $\{d(u_{i},v_{i})\}$ has limit $0$. (Right idea, but I'm not sure what a segment $\{[u_{i},v_{i}]\}$means in this space. You could stand to define the pair $u_{i},v_{i}$ by recursion to show explicitly what's going on here. --Steven.clontz 20:00, March 17, 2011 (UTC)) Now consider the two sequences $\{u_{i}\}$ and $\{v_{i}\}$. Both sequences converge, and they have to converge to the same point, else $\{d(u_{i},v_{i})\}$ doesn't converge to $0$. (I agree, but I'm not sure it's clear why? --Steven.clontz 20:00, March 17, 2011 (UTC)) Let $x$ be the limit point.

By construction, every $\epsilon$-neighborhood around $x$ contains points not in $V$ on one side, and points not in $U$ on the other side. (Could use clearer explanation why this is true. I wouldn't use the word "side" since there could be infinitely many points from U or V on either side of x. --Steven.clontz 20:00, March 17, 2011 (UTC)) Since $U$ and $V$ are both open, $x$ can't be in either set, since in an open set, sufficiently small $\epsilon$-neighborhoods can't contain points not in the set. But $U$ and $V$ are complements, so $x$ has to be in one or the other. This contradiction means that our assumption is false, and $X$ and $\emptyset$ are the only clopen sets in our metric space.

Olomana 02:47, March 16, 2011 (UTC)

This is the solution, but there are a few technical issues I had with it. Great idea, though! --Steven.clontz 20:00, March 17, 2011 (UTC)

I've been looking this over, and I can clear up some of the technical problems. However, I'm still asserting that a sequence converges without specifying the limit. This is something that I know to be true about the reals, but it's not something that we've already established in this class. So far, all we have is a definition of convergence to a given limit.

Olomana 17:44, March 18, 2011 (UTC)