Topology on real projective plane Crossley in his Essential Topology on page 27 introduces $\mathbb{R}P^2$. The set of all lines that pass through the origion which is also called the real projective plane.
He shows that the trivial definition of the open sets that correspond union of lines in $\mathbb{R}P^2$ to open subset of points in $\mathbb{R}^3$ will result in the indiscrete topology which means only the empty set and the whole set  of  $\mathbb{R}P^2$ are open. 
He suggests to avoid this we need only to omit the origin, the point $0$. 
But this doesn't make sense to me because even if we remove zero then we have again the indiscrete topology if we define open subsets in the real projective plane as those  union of lines that correspond to open subsets of points in $\mathbb{R}^3$
Suppose we have only the line $x$ in $\mathbb{R}P^2$ then any point on it, say 1, 2 or 3 doesn't have an open ball around it in $\mathbb{R}^3$ even if we omit 0.
My question is that why removing the origin from the real projective plane and a new definition for open sets changes the topology of the real projective plane from an indiscrete one to another new topology just like $\mathbb{R}^3-0$
 A: $\newcommand{\Reals}{\mathbf{R}}$Just so this has an answer: Fix an integer $n > 1$, let $\{\ell_{\alpha}\}_{\alpha \in I}$ be a family of lines through the origin in $\Reals^{n+1}$, and let $S^{n} \subset \Reals^{n+1}$ be the unit $n$-sphere centered at the origin.
If the union of the lines in the family
$$
U = \bigcup_{\alpha \in I} \ell_{\alpha} \subset \Reals^{n+1}
$$
is an open set, then in particular $U$ contains an open ball around the origin, so by homogeneity $U$ contains the open ball of radius $2$ (say) about the origin, so $S^{n} \subset U$, and thus $U = \Reals^{n+1}$.
By contrast, if $\ell' = \ell \setminus\{0\}$ is the complement of the origin in a line $\ell$, and if
$$
U' = \bigcup_{\alpha \in I} \ell'_{\alpha} \subset \Reals^{n+1} \setminus\{0\}
$$
is open, all we learn is that the set $U' \cap S^{n}$ is open in $S^{n}$. As wojowu says, there are many proper open subsets of $S^{n}$, such as complements of great hyperspheres (i.e., complemenets of hyperplanes intersected with the sphere), or open hemispheres.
It may help to consider $\Reals^{2}$: The only family of lines-through-the-origin containing a neighborhood of the origin is the family of all lines through the origin, but there are many, many families of lines-through-the-origin whose union is open in the punctured plane $\Reals^{2} \setminus\{0\}$.
