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

Open neighborhood of the Niemytzki plane, tangent to the x-axis

Open neighborhood of the Niemytzki plane, tangent to the x-axis

If $\mathrm{\Gamma }$ is the (closed) upper half-plane $\mathrm{\Gamma }=\{(x,y)\in {\mathbb{ℝ}}^2|y\ge 0\}$, then a topology may be defined on $\mathrm{\Gamma }$ by taking a local basis ${ℬ}(p,q)$ 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 $\mathrm{\Gamma }$.
• 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

${ℬ}(p,q)=\{\begin{array}{ll}\{U_{ϵ}(p,q):=\{(x,y):(x-p{)}^2+(y-q{)}^2<{ϵ}^2\}\mid ϵ>0\},& \text{if }q>0;\\ \{V_{ϵ}(p):=\{(p,0)\}\cup \{(x,y):(x-p{)}^2+(y-ϵ{)}^2<{ϵ}^2\}\mid ϵ>0\},& \text{if }q=0.\end{array}$

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

File:Moore plane, Nemytskii's tangent disk topology - topological space representation by open Neighbourhood.jpg

Moore Plane graphic representation

Properties

• The Moore plane $\mathrm{\Gamma }$ 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 $\{(x,0)\in \mathrm{\Gamma }|x\in R\}$ of $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ 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 $S:=\{(p,q)\in \mathbb{\mathbb{Q}}\times \mathbb{\mathbb{Q}}:q>0\}$ of points with rational coordinates is dense in $\mathrm{\Gamma }$; hence every continuous function $f:\mathrm{\Gamma }\to \mathbb{ℝ}$ is determined by its restriction to $S$, so there can be at most $|\mathbb{ℝ}{|}^{|S|}=2^{{\mathrm{\aleph }}_0}$ many continuous real-valued functions on $\mathrm{\Gamma }$.
2. On the other hand, the real line $L:=\{(p,0):p\in \mathbb{ℝ}\}$ is a closed discrete subspace of $\mathrm{\Gamma }$ with $2^{{\mathrm{\aleph }}_0}$ many points. So there are $2^{2^{{\mathrm{\aleph }}_0}}>2^{{\mathrm{\aleph }}_0}$ many continuous functions from L to $\mathbb{ℝ}$. Not all these functions can be extended to continuous functions on $\mathrm{\Gamma }$.
3. Hence $\mathrm{\Gamma }$ 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.