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 $$\mathbb{R}^2$$.

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)