In this question we consider the standard inclusion of $\mathbb{C}P^{n}\subset \mathbb{C}P^{n+1}$.

It is well known that $\mathbb{C}P^n$ is not a retract of $\mathbb{C}P^{n+1}$.

What about if we remove a finite subset as follows:

Question: Assume that $K\subset \mathbb{C}P^n$ is a finite set. Can $\mathbb{C}P^n\setminus K$ be a retract of $\mathbb{C}P^{n+1}\setminus K$?