# 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 is the (closed) 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*.

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

- The Moore 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.
- 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

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.

## References

- Stephen Willard.
*General Topology*, (1970) Addison-Wesley ISBN 0-201-08707-3. - Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978],
*Counterexamples in Topology*(Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446*(Example 82)* - "Niemytzki plane".
*PlanetMath*.