Let \(A\) be a proper subset of \(X\), and let \(B\) be a proper subset of \(Y\). If \(X\) and \(Y\) are connected, show that
\[ (X \times Y) - (A \times B) \]
is connected.
Solution
Let \(Z=(X\times Y)-(A\times B)\), and choose \(x_0\in X-A\) and \(y_0\in Y-B\). A union of connected subspaces with a point in common is connected.
1. For \(x\in X-A\) the slice \(\{x\}\times Y\), and for \(y\in Y-B\) the slice \(X\times\{y\}\), is a connected subset of \(Z\).
The slices are homeomorphic to \(Y\) and \(X\) by E-OTJ9S, hence connected. If \(x\notin A\), no point \((x,y)\) lies in \(A\times B\); similarly if \(y\notin B\).
2. The cross \(C_0=(\{x_0\}\times Y)\cup(X\times\{y_0\})\) is a connected subset of \(Z\).
By step 1, both slices are connected subsets of \(Z\), and they share \((x_0,y_0)\).
3. Every point \((x,y)\in Z\) lies in a connected subset of \(Z\) that contains \(C_0\).
1.If \(y\notin B\), the slice \(X\times\{y\}\) meets \(C_0\) at \((x_0,y)\), so \(C_0\cup(X\times\{y\})\) is connected.
4. Q.E.D.
By step 3, \(Z\) is a union of connected subsets that all contain \((x_0,y_0)\), so \(Z\) is connected.