# 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 $\Gamma$ is the (closed) upper half-plane $\Gamma =\{(x,y)\in \mathbb {R} ^{2}|y\geq 0\}$ , 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 $\Gamma \backslash \{(x,0)|x\in \mathbb {R} \}$ 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 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):

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 M; hence every continuous function $f:M\to \mathbb {R}$ 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 M.
2. On the other hand, the real line $L:=\{(p,0):p\in \mathbb {R} \}$ is a closed discrete subspace of M 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 M.
3. 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.