a subspace of $\mathbb R^3$ with $\pi_1=\mathbb Z_2$

Question

I've been wondering about such problems.

1. It is well known that $\mathbb{RP}^2$ cannot be realized as a subspace of $\mathbb R^3$.

But does there exist a space $X\subset\mathbb R^3$ (maybe even $CW$-complex) with $\pi_1(X)=\mathbb Z_2$?

2. If it is possible, can we find such $X$ to be homotopy equivalent to $\mathbb{RP}^2$?

I think that the first questions seems to be not so hard to answer but my knowledge is not enough to answer it myself.