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I have to learn for a very important test on monday but I don't get along with the following exercise from Bredon's book »Topology and Geometry«:

Let the projective plane $P^2=U_1\cup … \cup U_n$ where each $U_i$ is homeomorphic to the plane. Put $V_i=U_1\cup … \cup U_i$ for $1\leq i$U_i\cap V_{i-1}$ is disconnected or empty.

Any hints or proofs?

Thank you very very much!

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    $P^2$ is the projective plane? Why the abstract algebra and differential topology tags?2011-11-26
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    Your title has no relation to the actual question. It is of course true that there are empty sets in $P^2$ and there are also disconnected sets there...2011-11-26

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