Let $C\subset \mathbb P^n$ be a closed irreducible subset of dimension $m$. Is it true that the preimage by $q:\mathbb A^{n+1}\setminus \{0\}\rightarrow \mathbb P^n$ has dimension $m+1$ ?

If so, how can I prove it?

The definition of the dimension of a topological space that I am given is the following:

If $Y$ is an irreducible topological space, then $\operatorname{dim}(Y)$ is the biggest integer $m$, if it exists, such that there is a chain $Y=Y_{m} \supsetneq Y_{m-1} \supsetneq \cdots \supsetneq Y_{0} \text { with } Y_{i} \subset Y$ irreducible and closed in $Y$ ; if such an integer does not exist then $\operatorname{dim}(Y)=\infty .$