2
$\begingroup$

Let $A \subset \mathbb{R}^2$ be countable. Then it is not too hard to show that $\mathbb{R}^2 \setminus A$ is path-connected. However it is not always Manhattan connected since if $A = \mathbb{Q}^2 \setminus \{(0,0)\}$, the origin cannot be connected to any other point by a path moving only horizontally or vertically.

Say a subset of $\mathbb{R}^2$ is skew-Manhattan connected if there exist two directions so that any two points in the set can be connected by a path which moves in only those two directions.

What I want to decide is whether $\mathbb{R}^2 \setminus A$ as above is always skew-Manhattan connected?

2 Answers 2

5

The set of directions of the lines that go through two of your points is countable, so there is a direction $d$ such that all lines parallel to it contain at most one point from $A$. That direction and its orthogonal $d^\perp$ work.

Indeed: there is a line with direction $d^\perp$ which does not intersect $A$. Since we can move along it freely, it is enough ---in order to show skew-Manhattan connectedness--- to show that: if $L$ is a line with direction $d$ which contains a point $p$ from $A$, then we can connect points on one side of $p$ to points in the other side. This is easy, using cardinality considerations.

  • 0
    Accepted Jyrki's since he beat you by a few seconds! Thanks for both answers, I'm glad this had such a nice simple solution2011-09-19
  • 1
    +1 @nolion: I'm not sure, but I think that Mariano actually beat me to it by a few seconds (a message came while I was trying to click at the right place). You're welcome to change the accepted solution, if that was your criterion! May be a system time to the second can be found somehow?2011-09-19
  • 0
    Oops, I was just going off the order on the page, I guess you are a more reliable source. In general I don't think the quickest answer is a good way to decide but in this case you both said the same thing2011-09-19
  • 0
    @Jyrki: If you hover your mouse pointer over the date-and-time mark of a post, a tooltip that displays the date and the time (UTC) up to the second of when that post was committed shows up. In the case of this answer by Mariano, the time is `05:37:12Z`.2011-09-22
  • 0
    @J.M. Thank you very much. A nice feature.2011-09-22
3

The set $A$ is countable. Therefore so is the set of lines connecting two points of $A$. Theses lines have a countable set of slopes (counting infinity, if necessary), so it is easy to select two directions such that on any line parallel to either there is at most a single point in $A$. You can even choose the two directions to be orthogonal.