In mathematics, the Prüfer manifold or Prüfer surface is a 2-dimensional Hausdorff

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

 real analytic manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

 that is not paracompact

Paracompact space

In mathematics, a paracompact space is a topological space in which every open cover admits a locally finite open refinement. Paracompact spaces are sometimes also required to be Hausdorff. Paracompact spaces were introduced by ....

. It was introduced by and named after Heinz Prüfer

Heinz Prüfer

Ernst Paul Heinz Prüfer was a German mathematician, who worked on abelian groups, algebraic numbers, knot theory and Sturm-Liouville theory. His advisor was Issai Schur.- See also :* Prüfer manifold...

.

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 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 is the Prüfer manifold. The images of the points (0,0) in the spaces X_a form an uncountable discrete subset.