Moore plane

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.
If formula_1 is the (closed) upper half-plane formula_2, then a topology may be defined on formula_1 by taking a local basis formula_4 as follows:
That is, the local basis is given by
Thus the subspace topology inherited by formula_11 is the same as the subspace topology inherited from the standard topology of the Euclidean plane.
Proof that the Moore plane is not normal.
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):
In fact, if "X" is a separable topological space having an uncountable closed discrete subspace, "X" cannot be normal.