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 In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
. It was introduced by and named after Heinz Prüfer.


Construction

The Prüfer manifold can be constructed as follows . 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 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) In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither L ...


References

* * * {{DEFAULTSORT:Prufer Manifold Topological spaces Surfaces