Moore plane
In mathematics, the Moore plane, also sometimes called Niemytzki plane, is a topological space. It is a completely regular Hausdorff 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 is the 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.
Thus the subspace topology inherited by is the same as the subspace topology inherited from the standard topology of the Euclidean plane.
Properties
- The Moore plane is separable, that is, it has a countable dense subset.
- The Moore plane is a completely regular Hausdorff space, which is not normal.
- The subspace of 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.
Proof that the Moore plane is not normal
- 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.
- 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 .
- 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.