Moore plane on arvau

Wikipedia / 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 (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.

, then a topology by taking a local basis as follows:

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

is separable, that is, it has a countable dense subset.
The Moore plane is a completely regular Hausdorff space (i.e. Tychonoff space), which is not normal.
has, as its subspace topology, the discrete topology. Thus, the Moore plane shows that a subspace of a separable space need not be separable.
The Moore plane is first countable, but not second countable or Lindelöf.
The Moore plane is not locally compact.
The Moore plane is countably metacompact but not metacompact.

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):

.
many continuous functions from L.
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.