Show that the quotient space is homeomorphic to the n-disc
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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?
general-topology
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edited Dec 11 '18 at 13:10
Glorfindel
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