In mathematics, the Prüfer manifold or Prüfer surface is a 2-dimensional
Hausdorff real analytic
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex a ...
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. It was introduced by and named after
Heinz Prüfer
Ernst Paul Heinz Prüfer (10 November 1896 – 7 April 1934) was a German Jewish mathematician born in Wilhelmshaven. His major contributions were on abelian groups, graph theory, algebraic numbers, knot theory and Sturm–Liouville theory.
In ...
.
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)
References
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{{DEFAULTSORT:Prufer Manifold
Topological spaces
Surfaces