In mathematics, the Prüfer manifold or Prüfer surface is a 2-dimensional
Hausdorff real analytic
In mathematics, an analytic function is a function (mathematics), function that is locally given by a convergent series, convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are ...
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
that is not
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...
. 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
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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 sense "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither ...
References
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{{DEFAULTSORT:Prufer Manifold
Topological spaces
Surfaces
Eponyms in geometry