Let $S=\{\langle x,y,z\rangle:x^2+y^2+z^2=1\}$; this is the surface of the sphere of radius $1$ centred at the origin. What points must be removed from $S$ to get $P$? You have to remove the points the points $\langle x,y,z\rangle\in S$ such that $x^2+y^2=0$. Which points are these?
If $\langle x,y,z\rangle\in S$, then $x^2+y^2+z^2=1$, so if in addition $x^2+y^2=0$, it must be that $z^2=1$, and hence $z=\pm 1$. That is, the only points of $S$ that are removed to get $P$ are $\langle 0,0,1\rangle$ and $\langle 0,0,-1\rangle$. You could think of these two points as the north and south poles of $S$; $P$ is then everything except these two poles. You should have little trouble showing that $P$ is connected.
For example, notice that if $p$ and $q$ are distinct points of $P$, you can travel from $p$ to $q$ without leaving $P$: just follow a line of constant longitude from $p$ to the equator, then go round the equator until you reach the longitude of $q$, and finally follow a line of constant longitude from the equator to $q$.