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

• Elements of the local basis at points ${\displaystyle (x,y)}$ with ${\displaystyle y>0}$ are the open discs in the plane which are small enough to lie within ${\displaystyle \Gamma }$.
• Elements of the local basis at points ${\displaystyle p=(x,0)}$ are sets ${\displaystyle \{p\}\cup A}$ 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

${\displaystyle {\mathcal {B}}(p,q)={\begin{cases}\{U_{\epsilon }(p,q):=\{(x,y):(x-p)^{2}+(y-q)^{2}<\epsilon ^{2}\}\mid \epsilon >0\},&{\mbox{if }}q>0;\\\{V_{\epsilon }(p):=\{(p,0)\}\cup \{(x,y):(x-p)^{2}+(y-\epsilon )^{2}<\epsilon ^{2}\}\mid \epsilon >0\},&{\mbox{if }}q=0.\end{cases}}}$

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

Properties

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

See also

• Hedgehog space

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.