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

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

Properties

Proof that the Moore plane is not normal

The fact that this space \Gamma 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): In fact, if X is a separable topological space having an uncountable closed discrete subspace, X cannot be normal.