Prüfer manifold

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

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.

Construction

The Prüfer manifold can be constructed as follows: take an uncountable number of copies $(X_a{)}_{a\in \mathbb{ℝ}}$ of the plane, and take a copy $H$ of the upper half-plane. Then glue the open upper half of each plane $X_a$ to the upper half plane $H$ by identifying $(x,y)\in X_a$ 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 (2): 101–121. https://webhomes.maths.ed.ac.uk/~v1ranick/papers/rado.pdf. 

• 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 (1999). A Comprehensive Introduction to Differential Geometry. 1 (3rd ed.). Houston: Publish or Perish. ISBN 9780914098706.