Can we divide $\mathbb{R}^2$ into two connected parts such that each part is not simply-connected?

Can we divide $\mathbb{R}^2$ into two connected parts such that each part is not simply-connected?

My attempt

Put $A= \{ (0,0) \} $ and $B$ is the punctured plane.

Since that $S^1$ is a deformation retract of the punctured plane, $B$ is not simply-connected. Thus we can find a division of $\mathbb{R}^2$ such that one part is simply-connected but the other is not.

But how to deal with the problem above which requires that each part is not simply-connected?

It seems to be related to contractible and holes. But I don't know how to convert these ideas into precise mathematical language.

Any hints? Thanks in advance!

Added:

As pointed out in the comment, the counterexample exists.

Now I want to ask another question

Can we divide $\mathbb{R}^2$ into two path connected parts such that each part is not simply connected?

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