Why RP^1 is homeomorphic to S^1 ??

From: bigli <nimbigli@xxxxxxxxx>
Date: Fri, 05 Oct 2007 20:57:54 -0700

why RP^1 is homeomorphic to S^1 ?(please with details)

S^1 ={ (x,y) ; x^2 + y^2 =1 } together with induced topology of R^2 or
complex plane

RP^1 ={ { (x,y),(-x,-y) } ; (x,y) is in S^1 } together with quotient
topology induced by f: S^1 -------> RP^1 ; f(x,y)= { (x,y) ,(-x,-y) }

.

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