I had already posted this on mathoverflow and was advised to post the same here. So here it goes:
$X=\{(x,y,z)|x^2+y^2+z^2\le 1$ and $z≥0\}$ i.e. $X$ is the top half of a $3$-Disk.
$Z=X/E$, where $E$ is the equivalence relation on the the plane $z = 0$ which is as follows:
$(x,y,0)∼(−x,−y,0)$.
I was told that this space is equivalent to a cone of $\mathbb R\mathbb P^2$ (Real Projective Plane).
I want to know the following facts about "$Z$"
1) Is this a manifold with a boundary?
2) If it is a manifold with a boundary, what are the points of $Z$ that make the boundary?
3) Is it simply connected?
4) What is the minimum Euclidean dimension in which $Z$ can be embedded in?
Thank you very much for your help. I am new to topology and this problem came up as a part of my project. Any help is appreciated.
Thank you. Will.