# Show that the quotient space is homeomorphic to the n-disc

Let $$\sim$$ denote the equivalence relation on the $$n$$-sphere $$S^n$$ defined via $$(x_1,\dots,x_n,x_{n+1})\sim(x_1,\dots,x_n,−x_{n+1})\:\text{ for all }\: (x_1,\dots,x_{n+1})\in S^n.$$

Show that the quotient space $$S^n/\sim$$ is homeomorphic to the $$n$$-disc $$D^n$$.

I know that Ii have to show that there exists a bijective function $$f: (S^n/\sim) \to D^n$$ that is continuous and that the pre-image $$f^{-1}$$ is continuous as well, but I can't advance from here, any tips?

Hint: Define$$\begin{array}{rccc}f\colon&S^n/\sim&\longrightarrow&D^n\\&\bigl[(x_1,\ldots,x_n,x_{n+1})\bigr]&\mapsto&(x_1,\ldots,x_n).\end{array}$$
• Two things. First of all: it is not pre-image, it's the inverse function of $f$. Secondly: you asked for a tip and I provided one. – José Carlos Santos Dec 11 '18 at 13:44
• @MathMatich the inverse function is clear: map $(x_1, x_2,\ldots, x_n)$ to the class of $(x_1,x_2,\ldots,x_n, 1-\sqrt{\sum_{i=1}^n x^2_i})$. $f$ being continuous is clear (it's a projection in essence) and compactness does the rest as soon as you know it's a bijection. – Henno Brandsma Dec 11 '18 at 15:42
• I suppose that it's a typo and that it should be$$\left(x_1,x_2,\ldots,x_n,\sqrt{1-\sum_{n=1}^n{x_n}^2}\right).$$ – José Carlos Santos Dec 12 '18 at 18:04