Prüfer manifold

Short description: 2-d Hausdorff real analytic manifold that is not paracompact

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). Take an uncountable number of copies X_a 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 X_a to the upper half plane H by identifying (x,y)∈X_a 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 X_a under identification form an uncountable discrete subset.

See also

• Long line (topology)

References

• Radó, T. (1925), "Über den Begriff der Riemannschen Flächen", Acta Litt. Sci. Szeged 2: 101–121 
• Hazewinkel, Michiel, ed. (2001), "Prüfer surface", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/p075580 
• Spivak, Michael (1979), A comprehensive introduction to differential geometry. Vol. I (2nd ed.), Houston, TX: Publish or Perish, ISBN 978-0-914098-83-6, https://archive.org/details/comprehensiveint02spiv_0