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 [math]\displaystyle{ \Gamma }[/math] is the (closed) upper half-plane [math]\displaystyle{ \Gamma = \{(x,y)\in\R^2 | y \geq 0 \} }[/math], then a topology may be defined on [math]\displaystyle{ \Gamma }[/math] by taking a local basis [math]\displaystyle{ \mathcal{B}(p,q) }[/math] as follows:
- Elements of the local basis at points [math]\displaystyle{ (x,y) }[/math] with [math]\displaystyle{ y\gt 0 }[/math] are the open discs in the plane which are small enough to lie within [math]\displaystyle{ \Gamma }[/math].
- Elements of the local basis at points [math]\displaystyle{ p = (x,0) }[/math] are sets [math]\displaystyle{ \{p\}\cup A }[/math] 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
- [math]\displaystyle{ \mathcal{B}(p,q) = \begin{cases} \{ U_{\epsilon}(p,q):= \{(x,y): (x-p)^2+(y-q)^2 \lt \epsilon^2 \} \mid \epsilon \gt 0\}, & \mbox{if } q \gt 0; \\ \{ V_{\epsilon}(p):= \{(p,0)\} \cup \{(x,y): (x-p)^2+(y-\epsilon)^2 \lt \epsilon^2 \} \mid \epsilon \gt 0\}, & \mbox{if } q = 0. \end{cases} }[/math]
Thus the subspace topology inherited by [math]\displaystyle{ \Gamma\backslash \{(x,0) | x \in \R\} }[/math] is the same as the subspace topology inherited from the standard topology of the Euclidean plane.
Properties
- The Moore plane [math]\displaystyle{ \Gamma }[/math] 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 [math]\displaystyle{ \{(x,0)\in \Gamma | x\in R \} }[/math] of [math]\displaystyle{ \Gamma }[/math] 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 [math]\displaystyle{ \Gamma }[/math] 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 [math]\displaystyle{ S:=\{(p,q) \in \mathbb Q\times \mathbb Q: q\gt 0\} }[/math] of points with rational coordinates is dense in [math]\displaystyle{ \Gamma }[/math]; hence every continuous function [math]\displaystyle{ f:\Gamma \to \mathbb R }[/math] is determined by its restriction to [math]\displaystyle{ S }[/math], so there can be at most [math]\displaystyle{ |\mathbb R|^{|S|} = 2^{\aleph_0} }[/math] many continuous real-valued functions on [math]\displaystyle{ \Gamma }[/math].
- On the other hand, the real line [math]\displaystyle{ L:=\{(p,0): p\in \mathbb R\} }[/math] is a closed discrete subspace of [math]\displaystyle{ \Gamma }[/math] with [math]\displaystyle{ 2^{\aleph_0} }[/math] many points. So there are [math]\displaystyle{ 2^{2^{\aleph_0}} \gt 2^{\aleph_0} }[/math] many continuous functions from L to [math]\displaystyle{ \mathbb R }[/math]. Not all these functions can be extended to continuous functions on [math]\displaystyle{ \Gamma }[/math].
- Hence [math]\displaystyle{ \Gamma }[/math] 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
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 (Example 82)
- "Niemytzki plane". http://planetmath.org/?op=getobj&from=objects&id={{{id}}}.
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