isomorphic fundamental groups of quotient space

Question

I have a set $X\subset \mathbb R^n$ an equivalence relation $x∼-x$. Say $Y\cong X$ i.e they are homeomorphic. I would like to conclude $\pi_1(X/∼)=\pi_1(Y/∼)$.

Is that true? It does look reasonable. problem is that as far as I know continuous maps only give homomorphism of fundamental groups.

Thanks

Answer 1

Take the sphere $\mathbb S^{n-1}$ and a "translated sphere" $Y = \{x + 1 \mid x \in \mathbb S^{n-1}\}$. We have $(Y/\sim) \cong Y$ and $\mathbb (S^{n-1}/\sim) \cong \mathbb R \mathbb P^{n-1}$ which has different fundamental group than $Y$.