Give an example of a connected closed subset $C$ of $\mathbb{R}^2$ such that $\mathbb{R}^2 \setminus C $ Give an example of a connected closed subset $C$ of $\mathbb{R}^2$ such that $\mathbb{R}^2 \setminus C $ has infinitely many components. 

can somebody help me please.thanks for your time
 A: HINT: Think of the lines dividing up an infinite checkerboard:
$$\begin{array}{c|c|c|c}
\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot\\ \hline
\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot\\ \hline\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot\\ \hline\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot\\ \hline\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot\\ \hline\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot\\ \hline\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot&\color{white}\cdot
\end{array}$$
