Has $X\times X$ the following property? Let $X$ be a topological space that satisfies the following condition at each point $x$:
For every open set $U$ containing $x$, there exists an open set $V$ with compact boundary such that $x\in V\subseteq U$.
Does $X\times X$ also have that property?

Try to Proof: For every open set $W$ of $X \times X$, there exists $U_1$ and $U_2$ of $X$ such that $U_1 \times U_2 \subset W.$ Then there exist $V_1$ and $V_2$ such that there compact boundary are contained in $U_1$ and $U_2$ respectively.
Next we prove that the compact boundary of $V_1 \times V_2$ are contained in $U_1\times U_2$. In fact, $Fr (V_1 \times V_2)= \overline{V_1 \times V_2} \setminus (V_1 \times V_2)=\overline{V_1} \times \overline{V_2} \setminus (V_1 \times V_2)=((\overline{V_1}\setminus V_1)\times \overline{V_2}) \cup (\overline{V_1 }\times (\overline{V_2}\setminus V_2))=(Fr(V_1) \times \overline{V_2}) \cup  (\overline{V_1 }\times Fr(V_2))$. 
We cannot be sure that $(\overline{V_1}\setminus V_1)\times \overline{V_2}$ or $\overline{V_1 }\times (\overline{V_2}\setminus V_2)$ is compact unless $V_1$ and $V_2$ are compact.
