A set is connected if and only if it cannot be the union of two non-enpty separarted sets.
That is two set neither of which contains a point nor a limit point of the other.
Is there a simple test for connectedness for subsets of R^n? For example, to test if such subsets are compact, you can just check if they are closed and bounded in R^n.
I was wondering if there was such a test for connectedness, as I'm not sure how to decide whether arbitrary sets in R^n are connected. For example, (x,y) in R^2 where at least one of x and y is rational.
Any help would be greatly appreciated
Thanks, but I'm still not sure how to see this for arbitrary sets. For example for the set I posted in the OP, could you tell me whether it is connected or not and why?
I was just hoping there was a check for connectness that I find easy, like the check for compactness I mentioned.
Well I find this is the easiest method:
Like for example the set you talked about, lets call it .A set is connected if and only if it cannot be the union of two non-enpty separarted sets.
That is two set neither of which contains a point nor a limit point of the other.
If it is not connected then there are two open sets
that divide the set.
Now for every vertical line with a rational coordinate, then every point on that line is in
and every point on that line must be in the same set or . Do you see why?
Now the horizontal lines are the same, for every horizontal line with a rational y-coordinate, the whole line is in or .
Now pick the vertical line e.g.
and lets assume that it is in , do you see how that leads to every point in the set is in
so that is connected?
Thanks for the help, that makes sense. As it happens I found a condition for disconnectedness in my notes that makes it a lot easier to me. The statement is: a set T is disconnected if and only if there exists a subset of T (which is not itself or the empty set) that is both open and closed.
For example, for the set I mentioned, this makes it really easy to check. Any non-trivial subset cannot be open (or closed in fact) as for any x in the set, a ball centred at x will contain an infinite amount of points where both coordinates are irrational. Hence the set cannot be disconnected.