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.

Definition
If $$\Gamma$$ is the (closed) upper half-plane $$\Gamma = \{(x,y)\in\R^2 | y \geq 0 \}$$, then a topology may be defined on $$\Gamma$$ by taking a local basis $$\mathcal{B}(p,q)$$ as follows:


 * Elements of the local basis at points $$(x,y)$$ with $$y>0$$ are the open discs in the plane which are small enough to lie within $$\Gamma$$.
 * Elements of the local basis at points $$p = (x,0)$$ are sets $$\{p\}\cup A$$ 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
 * $$\mathcal{B}(p,q) = \begin{cases} \{ U_{\epsilon}(p,q):= \{(x,y): (x-p)^2+(y-q)^2 < \epsilon^2 \} \mid \epsilon > 0\}, & \mbox{if }  q > 0;  \\ \{ V_{\epsilon}(p):= \{(p,0)\} \cup \{(x,y):  (x-p)^2+(y-\epsilon)^2 < \epsilon^2 \} \mid \epsilon > 0\},  & \mbox{if } q = 0. \end{cases} $$

Thus the subspace topology inherited by $$\Gamma\backslash \{(x,0) | x \in \R\}$$ is the same as the subspace topology inherited from the standard topology of the Euclidean plane.



Properties

 * The Moore plane $$\Gamma$$ 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 $$\{(x,0)\in \Gamma | x\in R \}$$ of $$\Gamma$$ 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 $$\Gamma$$ 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):
 * 1) On the one hand, the countable set $$S:=\{(p,q) \in \mathbb Q\times \mathbb Q: q>0\}$$ of points with rational coordinates is dense in $$\Gamma$$; hence every continuous function $$f:\Gamma \to \mathbb R$$ is determined by its restriction to $$S$$, so there can be at most $$|\mathbb R|^{|S|} = 2^{\aleph_0}$$ many continuous real-valued functions on $$\Gamma$$.
 * 2) On the other hand, the real line $$L:=\{(p,0): p\in \mathbb R\}$$ is a closed discrete subspace of $$\Gamma$$ with $$ 2^{\aleph_0}$$ many points. So there are $$2^{2^{\aleph_0}} > 2^{\aleph_0}$$ many continuous functions from L to $$\mathbb R$$. Not all these functions can be extended to continuous functions on  $$\Gamma$$.
 * 3) Hence $$\Gamma$$ 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.