# Proper subsets of connected spaces and proper product

Let $$A$$ be a proper subset of $$X$$ and $$B$$ a proper subset of $$Y$$. If $$X,Y$$ are connected. Show that

$$X\times Y\backslash (A\times B)$$ is connected.

Lemma: Let $$X$$ be a space and $$A_1,A_2...,A_n$$ a finite sequence of connected subsets in $$X$$. If $$A_j\cap A_{j+1}\neq \varnothing$$ for each $$j=1,2...,n-1$$ then $$A_1\cup A_2 \cup...A_n$$ is connected.

My attempt:

Fix $$a\in X\backslash A$$ and fix $$b\in Y\backslash B$$..

Observe, $$\{$$ $$a$$ $$\} \times Y$$ and $$X\times \{$$ b $$\}$$ are connected. Let $$x\in X\backslash A$$ and$$y\in Y\backslash B$$. Put

$$T_{xy}=(\{x \}\times Y)$$ $$\cup (X\times \{b\}) \cup (\{a\} \times Y$$) $$\cup$$ $$(X \times \{y \})$$ Then, $$T_{xy}$$ is connected because of the above lemma. Put $$T=\bigcup_{x\in X\backslash A}\bigcup_{y\in Y\backslash B} T_{xy}$$ . Then, $$T$$ is connected because each $$T_{xy}$$ contains $$(a,b)$$ and each $$T_{xy}$$ is connected

Is this correct?

The lemma which applies here would be : Let, $$X$$ be a topological space and $$\{A_{\alpha}\}$$ be a collection of connected subsets of $$X$$ such that $$A_{\alpha} \cap A_{\beta} \neq \emptyset \forall \alpha, \beta$$ then $$\cup_{\alpha} A_{\alpha}$$ is connected.
• The finite many sets are $\{$ $x$ $\}$ $\times Y$, $X\times \{b\}$, $\{a \} \times Y$, $X\times \{y\}$. I'm invoking the lemma on these sets Call $A_1=\{ x\} \times Y$ and ... Apr 28 '20 at 1:45
• Yes but then you are taking union of all the $T_{xy}$ to conclude the result. I am saying that the result may not be applicable there. Apr 28 '20 at 1:49
• You can check on this space: $\Bbb R \times \Bbb R - \Bbb Q \times \Bbb Q$. Apr 28 '20 at 1:50