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) which is not normal. 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:
- At points with , the open neighbourhoods are the open discs in the plane which are small enough to lie within . This is just the subspace topology of the usual topology of the Euclidean plane.
- At points , the open neighbourhoods 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
Properties
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):
- 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.
- 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.
- 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
References