Is it true that $RP^3$ minus a point deformation retracts to a space homeomorphic to $RP^2$

Question

I have the following question:

Is it true that $RP^3$ minus a point deformation retracts to a space homeomorphic to $RP^2$

My Efforts

Answer is NO

Assume on the contrary that $RP^3$ with a point deleted deformation retracts onto a space homeomorphic to $RP^2$. It implies that fundamental group of $RP^3$ minus a point is $\mathbb{Z}/2\mathbb{Z}$

Deleting a point from $RP^3$ is same as removing $2$ points from $S^3$. By stereographic projection $S^3$ minus a point is homeomorphic to $\mathbb{R}^3$. As $R^3$ minus a point is simply connected, it's fundamental group is trivial. So we arrive at a contradiction.

Answer 1

Hint: Assume you have removed $[0,0,0,1]$. Consider the map defined by $f_u([x,y,z,t])=([x,y,z,tu]), u\in [0,1]$.

Answer 2

$\mathbb R\mathrm P^3$ has a CW structure given by attaching a $3$-cell to $\mathbb R\mathrm P^2$. We can arrange for the puncture to be within the interior of the $3$-cell, and a punctured $3$-cell deformation retracts onto its boundary.