Hello I have problems with this exercise
Let $X, Y$ be two homeomorphic topological spaces. Prove $X$ is Hausdorff if and only if $Y$ is Hausdorff
My attempt:
Let $ f: X \to Y $ be the homeomorphism between the two and $ X $ is Hausdorff, to see that $ Y $ is also Hausdorff, let's take two points $ y, y^\prime\in Y $ different let's take it to $ X $ by $ f^{-1} $; How can I separate them for open? (because $ X $ is Hausdorff) and transfer them to open that separate $y,y^\prime$ using $ f $.
How would it be if $Y$ is Hausdorff?
Take $x,x'\in X$ such that $f(x)=y$ and that $f(x')=y'$. Since $X$ is Hausdorff, there are neighborhoods $V$ of $x$ and $V'$ of $x'$ which do not intersect. Since $f^{-1}$ is continuous, $f(V)$ and $f(V')$ are neighborhoods of $y$ and of $y'$ respectively. But, since $V\cap V'=\emptyset$ and $f$ is a bijection, $f(V)\cap f(V')=\emptyset$ too.