The natural map $\text{Spec}A_f\to \text{Spec}A$ is a homeomorphism onto $D(f)$

Question

Let $A$ be a commutative ring with $1$ and $f\in A$ a non-nilpotent element. Then the set $\text{Spec}A_f$ has a natural one-to-one correspondence with the subset $D(f)\subset \text{Spec}A$ consisting of all $\mathfrak{p}\in \text{Spec}A$ with $f\notin \mathfrak{p}$. So $D(f)$ is an open subset of $\text{Spec}A$. Is it true that the natural map $\text{Spec}A_f\to \text{Spec}A$ is a homeomorphism onto $D(f)$? (I merely know that this map is a continuous injection onto $D(f)$.)