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(Solved): Problem 1. The real projective plane RP^(2) is the topological space of lines passing through the or ...
Problem 1. The real projective plane RP^(2) is the topological space of lines passing
through the origin in R^(3). More precisely, let RP^(2) be the set of equivalence classes x
of elements of R^(3) under the equivalence relation
AAx,yinR^(3),x?yLongleftrightarrowEE\lambda !=0 such that y=\lambda x
Let q:(R^(3))/(()/()){0}->R^(2) be the map q(x)=[x] and give RP^(2) the quotient topology
induced by this surjective map.
(a) Prove that RP^(2) is compact and connected.
(b) Prove that the map
f:RP^(2)->R
given by
f([(x_(1),x_(2),x_(3))])=(|x_(3)|)/(\sqrt(x_(1)^(2)+x_(2)^(2)+x_(3)^(2)))
is well-defined and continuous.
Correct answers to (a), (b) will be short, essentially referencing a few relevant facts. PLEASE PROVIDE FULL ANSWERS. THUMBS UP PROMISED