Prüfer manifold

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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.


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 (xy) with y > 0). Then glue 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 is the Prüfer manifold. The images of the points (0,0) in the spaces Xa form an uncountable discrete subset.

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