Prüfer manifold

In mathematics, the Prüfer manifold or Prüfer surface is a 2-dimensional Hausdorff real analytic manifold that is not paracompact. It was introduced by Radó (1925) and named after Heinz Prüfer.

Construction

The Prüfer manifold can be constructed as follows (Spivak 1979, appendix A). Take an uncountable number of copies Xa of the plane, one for each real number a, and take a copy H of the upper half plane (of pairs (x, y) with y > 0). Then glue the open upper half of each plane Xa to the upper half plane H by identifying (x,y)∈Xa for y > 0 with the point (a + yx, y) in H. The resulting quotient space Q is the Prüfer manifold. The images in Q of the points (0,0) of the spaces Xa under identification form an uncountable discrete subset.

See also

References
  • Radó, T. (1925), "Über den Begriff der Riemannschen Flächen", Acta Litt. Sci. Szeged, 2: 101–121
  • Solomentsev, E.D. (2001) [1994], "Prüfer surface", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  • Spivak, Michael (1979), A comprehensive introduction to differential geometry. Vol. I (2nd ed.), Houston, TX: Publish or Perish, ISBN 978-0-914098-83-6, MR 0532830
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