Let S be a projective plane and let m be any line in S. Define a set S, as follows. The points of S, are the points of S that are not on m. The lines in S, are the lines in S with points in common with m removed. (a) Prove that S, is an affine plane. (b) Prove that the completion of S, is isomorphic to S.
Hello, here's my attempt and thought process but I am stuck on the final steps of the proof.
Proof of a: [NTS: the S0 satisfies the axioms of the affine plane]
Axiom 1:Every pair of points lies on exactly one line. Since the points in S0 are the points in S not on m they must follow the axioms of the projective plane where each pair of points lie on exactly one line. Hence, every pair of points in S0 lie on exactly one line.
Axiom 2:Given a line l and a point P not on l, there exists exactly one line through P that is parallel to l Let l be a line in S0. Since the lines in S0 are the lines in S with the points in common with m removed, m contains a point P not on l. Hence, m and l are parallel since they do not share a point. Therefore l and m are in the same pencil of parallel lines and share an ideal point Q...
This is as far as I have gotten and I am still not sure that what I have so far is on the right track. My idea was to show that there cannot be another line through P parallel to l using the line at infinity but that doesn't sound correct.
Axiom 3: There exists four points, no three of which are collinear Since S is a projective plane, it must have four points A,B,C,D, no three of which are collinear. My thought process is that since no three of the points are collinear, at MOST 2 of the points are on line m. Then, I tried to think about proving this axiom with a proof by cases.. Case one: All of A,B,C,D are in S0 and not on m Case two: A,B,C are in S0 and D is on m Case three: A,B are in S0 and C,D are on m In each of those cases, I know I would need to show that there are four points in S0 now that no three of which are collinear... but after all of that I'm worried that my thought process is not correct because I am not sure how to prove cases two and three without simply introducing more points and just stating that they fit the axiom (as in, creating a pointless proof that says that if D is on m then there must be another point because there has to be four and its not collinear because it isn't supposed to be and if it were then I wouldn't be proving my point)
I want to prove that S0 is an affine plane by proving each axiom individually and thoroughly. If someone can provide their proof following this outline and explain each step in their thought process it will be greatly appreciated.