Determine which of the following subsets of R2 open, closed or neither 
Determine which of the following subsets of $\Bbb R^2$ are open, closed or neither:
  \begin{align}
&1.\quad A=\{(x,y)\in \Bbb R^2:x+y=1\}\\
&2. \quad A=\{(x,y)\in \Bbb R^2:  x+y>1\}\\
&3. \quad A=\{(x,y)\in \Bbb R^2:x,y\in \Bbb{Q}  \}\\
&4. \quad \Bbb{R}^2\setminus\{1,-1\}
\end{align}



*

*closed

*open

*closed

*open


if wrong, please correct me please
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The third set $\{(x,y):x,y\in\mathbb{Q}\}$ is neither open nor closed in $\mathbb{R}^2$, because $\mathbb{Q}$ is neither closed nor open in $\mathbb{R}$. Note that $\mathbb{Q}$ is dense in $\mathbb{R}$, that is every real number is a limit point of $\mathbb{Q}$, however the irrational numbers $\not\in\mathbb{Q}$. Hence $\mathbb{Q}$ is not closed in $\mathbb{R}$. Again take any rational $r\in\mathbb{Q}$. Then any interval $(r-\epsilon,r+\epsilon)\not\subset\mathbb{Q}$ because it contains irrational numbers. Hence $\mathbb{Q}$ is not open in $\mathbb{R}$.
