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