# Quotient map $f$ from the $n$-sphere to the $n$-disk [duplicate]

I'm trying to construct a quotient map $$f$$ from the $$n$$-sphere $$S^n$$ to the $$n$$-disk $$D^n$$, in which $$f(x_1,x_2,\ldots,x_n,x_{n+1})=f(x_1,x_2,\ldots,x_n,-x_{n+1})$$. The goal is to show that $$S^n/\sim_f$$ is homeomorphic to $$D^n$$. (Is this the right way to go about it?)
## marked as duplicate by Henno Brandsma general-topology StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Dec 11 '18 at 15:57
• If you're willing to go the other way, you can get the homeomorphism directly: $x\mapsto (2\sqrt {1-\|x\|^2}x,2\|x\|^2-1)$ – Matematleta Dec 11 '18 at 16:14