1. ## Connectivity

Let $\displaystyle A$ be a union of a countable number of "lines" in $\displaystyle \mathbb{R}^3$.
Prove that $\displaystyle B := \mathbb{R}^3 - A$ is connected.

This seems intuitively true - take two points $\displaystyle a, b \in B$. Look at the line $\displaystyle [a, b] = \{ ta + (1-t)b \ : \ t \in [0,1] \}$. If a point on the line intersects one of the lines in A, then simply deviate the connecting path a little bit, such that you don't intersect with the next closest line.

Making this rigorous, however, seems a bit painful, moreso considering that it's a question from a previous exam. Any ideas/hints are welcome.

Thanks, Defunkt.

2. Originally Posted by Defunkt
Let $\displaystyle A$ be a union of a countable number of "lines" in $\displaystyle \mathbb{R}^3$.
Prove that $\displaystyle B := \mathbb{R}^3 - A$ is connected.

This seems intuitively true - take two points $\displaystyle a, b \in B$. Look at the line $\displaystyle [a, b] = \{ ta + (1-t)b \ : \ t \in [0,1] \}$. If a point on the line intersects one of the lines in A, then simply deviate the connecting path a little bit, such that you don't intersect with the next closest line.

Making this rigorous, however, seems a bit painful, more so considering that it's a question from a previous exam. Any ideas/hints are welcome.
There are uncountably many planes containing the two points a and b. They are "almost disjoint", in the sense that the intersection of any two of them is just the line joining a to b. Since there are only countably many lines in A, at least one of these planes does not contain any of the lines.

Choose such a plane. It contains at most one point from each line in A. So that reduces the problem from three dimensions to two. The two-dimensional problem is this: Given a countable set of points B in a plane, show that the complement of B is connected.

The proof of that is a similar counting argument. Given two points a and b in the plane, there are uncountably many disjoint circular arcs connecting a and b, with the property that any two of them are almost disjoint (in the sense that their intersection consists only of the points a and b). Only countably many of these can contain a point in B. Choose an arc containing no such points. That demonstrates that the complement of B is connected (in fact, path-connected).

3. Thanks for the quick answer. I asked the prof. how to do it as well, at first he didn't come up with an answer but he proposed the same solution the next time I saw him.