Let $D^n\subset \mathbb{R}^n$ be the subset consisting of those points $(x_1,\dots,x_n)\in \mathbb{R}^n$ such that $x_1^2+\dots+x_n^2\leq 1$ and let $S^{n-1}\subset D^n$ be the subset of those points $(x_1,\dots,x_n)\in D^n$ such that $x_1^2+\dots+x_n^2=1$ (i.e. the boundary of $D^n$). Define an equivalence relation $\sim$ on $S^{n-1}\subset D^n$ by $$(x_1,x_2,\dots,x_n)\sim (-x_1,x_2,\dots,x_n).$$
Is the quotient $D^n/\sim$ homeomorphic to $S^n$? I think this is true for $n=2$, and am curious about higher dimensions.
This is true for all $n$: $(D^n/\sim)\cong S^n$.
To see this, first note that $\sum ( D^n/\sim) \cong D^{n+1}/\sim$ where $\sum$ denotes the suspension functor: $\sum X$ is $X\times [-1,1]$ with $X\times \{1\}$ and $X\times \{-1\}$ collapsed to points. One such homeomorphism is $f([x_1,..., x_n],t)\mapsto [(\sqrt{1-t^2}\,x_1,...,\sqrt{1-t^2}\,x_n,t)].$
Now, ($D^1/\sim) = S^1$ because in this case $\sim$ is the usual identification $S^1 \cong D^1/\partial D^1$. Now, the fact that $(D^n/\sim)\cong S^n$ follows from induction: $(D^{n+1}/\sim) \cong \sum(D^n/\sim)\cong \sum(S^n) \cong S^{n+1}$.