Let $X$ be the closed unit ball $\{ x^2 + y^2 \leq 1 \}$ in $\mathbb R^2$ and let $X^*$ be the partition of $X$ consisitingof all the one point set $\{ x \times y \}$ for which $x^2 + y^2 < 1$, along with the set $S^1 = \{ x^2 + y^2 = 1 \}$. Let $p : X \rightarrow X^*$ be the surjective map carry each point of $X$ to the element of $X^*$ containg it. A topology $\tau_p$ defined on $X^*$ induced by $p$ such that a subset $U$ of $X^*$ is open if $P^{-1}(U)$ is open in $X$ ; then $(X^*, \tau_p)$ is called Quotien Space of $X$ induced by $p$.
What are the saturated open sets of $X$, where saturated set is a subset of $X$ with respect to $p$ if it is the form $P^{-1}(U)$, where $U \subset X^*$
I think open saturated subsets of x are $\{ (x,y) | r < x^2 + y^2 \leq 1 \}$, where $ 0 < r <1$ and open balls contained in $X$.
what are the open subsets of $X^*$. Also show that $X^*$ is homeomorphic to the subspace $ S^2 = \{ (x,y,z) | x^2 + y^2 +z^2 = 1\}$ of $\mathbb R^3$
Any help would be appreciated. Thank you