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 of the plane, and take a copy of the upper half-plane. Then glue the open upper half of each plane to the upper half plane by identifying for with the point in . The resulting quotient space is the Prüfer manifold. The images in of the points of the spaces under identification form an uncountable discrete subset.
See also Long line (topology)
References