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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.
Contents
Definition
If
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
S := { ( p , q ) ∈ Q × Q : q > 0 } of points with rational coordinates is dense in M; hence every continuous functionf : M → R is determined by its restriction toS , so there can be at most| R | | S | = 2 ℵ 0 - On the other hand, the real line
L := { ( p , 0 ) : p ∈ R } is a closed discrete subspace of M with2 ℵ 0 2 2 ℵ 0 > 2 ℵ 0 R . 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.