Intuition behind the inverse image of quotient map in a topological vector space

Question

Let $Y$ be a closed subspace of a Topological vector space $X$. Let us now consider the quotient map $\pi:X \to X/Y$ defined by $$\pi(x):=x+Y,~~~\forall~x \in X.$$ In my lecture note it is given that $\pi^{-1}\left(\pi(x)\right)=Y+x.$ Also, if $U$ is an open set in $X$ then it is given that $\pi^{-1}\left(\pi(U)\right)=Y+U.$
I know that, $$\pi^{-1}(\pi(x))=\{z \in X ~:~\pi(z)=x+Y\}.$$ But I am totally confused with the above argument.
Thanks for your little time.

Answer 1

Well, $$z+Y=\pi(z)=x+Y\iff z-x\in Y\iff z\in x+Y\,.$$ Similarly, $$z\in\pi^{-1}(\pi(U))\iff \pi(z)=\pi(u)\,\text{ for some }u\in U\iff z\in u+Y\,\text{ for some }u\in U\iff z\in U+Y\,.$$