1
$\begingroup$

Let $X,Y$ be connected sets and $A,B$ proper subsets of $X$ and $Y$ respectively. Then it can be shown that the space $ (X \times Y) \setminus (A \times B)$ is connected. This is an problem from Munkres book. My question is:

Is the above result still true when we replace connectedness by path-connectedness instead?

1 Answers 1

4

Yes, and the proof is even easier than the connectedness proof. You just need to construct a path between any two points, which you can do piecewise, staying in subsets of the form $X\times\{y\}$ and $\{x\}\times Y$, which are path connected.

Alternatively you can copy the connectedness proof, which consists of covering $X\times Y\setminus A\times B$ in stages by connected sets which have a point in common. This still works for path connectivity.