$(X, \mathscr T_X)$ and $(Y, \mathscr T_Y)$ be 2 topological spaces. $Y$ be Hausdorff.
$f$ be a continuous function, $f: X \to Y$. To show that if $f$ is injective $\implies$ $X$ is Hausdorff.
Here's how I tried:
Let $f(x_1)$ and $f(x_2)$ be two elements of Y. Since $Y$ is Hausdorff we can find 2 disjoint neighborhoods around $f(x_1)$ and $f(x_2)$. Let it be $U, V$ respectively.
Since $f$ is continuous there exists an open neighborhood around $x_1$ and $x_2$ in X (let it be $E(x_1)$ and $ F(x_2)$) such that $f(E) \subset U$ and $f(F) \subset V$.
After this how to use injectivity property to prove that E is disjoint with F??
Best Answer
Let $x \neq y$ be elements of X,by injectivity you get $f(x)\neq f(y)$. Now pick a neighbourhood $U$ of $f(x)$ and $V$ for $f(y)$ such that $U \cap V=\emptyset$; $f^{-1}(U) \cap f^{-1}(V)=f^{-1}(U \cap V) = \emptyset$ and they are neighbourhoods of $x$ and $y$ respectively.