Moore plane

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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 (also called Tychonoff space) that is not normal. 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 . Thus the subspace topology inherited by is the same as the subspace topology inherited from the standard topology of the Euclidean plane.
  • 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

Moore Plane graphic representation

Properties[edit]

Proof that the Moore plane is not normal[edit]

The fact that this space M 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 M; hence every continuous function is determined by its restriction to , so there can be at most many continuous real-valued functions on M.
  2. On the other hand, the real line is a closed discrete subspace of M with many points. So there are many continuous functions from L to . Not all these functions can be extended to continuous functions on M.
  3. Hence M 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]