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.
gollark: Continue.
gollark: It's very useful when I need to debug some mess potatOS does.
gollark: PotatOS actually has global stack tracing built in.
gollark: Like <@!151391317740486657> .
gollark: Well, back when I was advertising it to new users, lots complained that they could not solve such a hard maths problem and stuff.
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", Encyclopedia of Mathematics, EMS Press
- 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
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.