Prüfer manifold

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In topology, the Prüfer manifold or Prüfer surface is a 2-dimensional Hausdorff real analytic manifold that is not paracompact. It was introduced by Tibor Radó in 1925 and named after Heinz Prüfer.[1]

Construction

The Prüfer manifold can be constructed as follows:[2] take an uncountable number of copies ( X a ) a ∈ R {\displaystyle (X_{a})_{a\in \mathbb {R} }} {\displaystyle (X_{a})_{a\in \mathbb {R} }} of the plane, and take a copy H {\displaystyle H} {\displaystyle H} of the upper half-plane. Then glue the open upper half of each plane X a {\displaystyle X_{a}} {\displaystyle X_{a}} to the upper half plane H {\displaystyle H} {\displaystyle H} by identifying ( x , y ) ∈ X a {\displaystyle (x,y)\in X_{a}} {\displaystyle (x,y)\in X_{a}} for y > 0 {\displaystyle y>0} {\displaystyle y>0} with the point ( a + y x , y ) {\displaystyle (a+yx,y)} {\displaystyle (a+yx,y)} in H {\displaystyle H} {\displaystyle H}. The resulting quotient space Q {\displaystyle Q} {\displaystyle Q} is the Prüfer manifold. The images in Q {\displaystyle Q} {\displaystyle Q} of the points ( 0 , 0 ) {\displaystyle (0,0)} {\displaystyle (0,0)} of the spaces X a {\displaystyle X_{a}} {\displaystyle X_{a}} under identification form an uncountable discrete subset.

See also

References

  1. Radó, T. (1925). "Über den Begriff der Riemannschen Flächen" (PDF). Acta Litt. Sci. Szeged. 2 (2): 101–121.
  2. Spivak (1999), appendix A.