In mathematics, the Moore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii's tangent disk topology), is a topological space. It is a completely regular Hausdorff space (that is, a Tychonoff space) that is not normal. It is an example of a Moore space that is not metrizable. It is named after Robert Lee Moore and Viktor Vladimirovich Nemytskii.

Definition

edit
 
Open neighborhood of the Niemytzki plane, tangent to the x-axis

If   is the (closed) upper half-plane  , then a topology may be defined on   by taking a local basis   as follows:

  • Elements of the local basis at points   with   are the open discs in the plane which are small enough to lie within  .
  • Elements of the local basis at points   are sets   where A is an open disc in the upper half-plane which is tangent to the x axis at p.

That is, the local basis is given by

 

Thus the subspace topology inherited by   is the same as the subspace topology inherited from the standard topology of the Euclidean plane.

 
Moore Plane graphic representation

Properties

edit

Proof that the Moore plane is not normal

edit

The fact that this space   is not normal can be established by the following counting argument (which is very similar to the argument that the Sorgenfrey plane is not normal):

  1. On the one hand, the countable set   of points with rational coordinates is dense in  ; hence every continuous function   is determined by its restriction to  , so there can be at most   many continuous real-valued functions on  .
  2. On the other hand, the real line   is a closed discrete subspace of   with   many points. So there are   many continuous functions from L to  . Not all these functions can be extended to continuous functions on  .
  3. Hence   is not normal, because by the Tietze extension theorem all continuous functions defined on a closed subspace of a normal space can be extended to a continuous function on the whole space.

In fact, if X is a separable topological space having an uncountable closed discrete subspace, X cannot be normal.

See also

edit

References

edit
  • Stephen Willard. General Topology, (1970) Addison-Wesley ISBN 0-201-08707-3.
  • Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446 (Example 82)
  • "Niemytzki plane". PlanetMath.