{{Short description|Topological space}} 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== right|Open neighborhood of the Niemytzki plane, tangent to the x-axis If <math>\Gamma</math> is the (closed) upper half-plane <math>\Gamma = \{(x,y)\in\R^2 | y \geq 0 \}</math>, then a topology may be defined on <math>\Gamma</math> by taking a local basis <math>\mathcal{B}(p,q)</math> as follows:
*Elements of the local basis at points <math>(x,y)</math> with <math>y>0</math> are the open discs in the plane which are small enough to lie within <math>\Gamma</math>. *Elements of the local basis at points <math>p = (x,0)</math> are sets <math>\{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>\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} </math>
Thus the subspace topology inherited by <math>\Gamma\backslash \{(x,0) | x \in \R\}</math> is the same as the subspace topology inherited from the standard topology of the Euclidean plane.
thumb|Moore Plane graphic representation
==Properties== *The Moore plane <math>\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>\{(x,0)\in \Gamma | x\in R \}</math> of <math>\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>\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>S:=\{(p,q) \in \mathbb Q\times \mathbb Q: q>0\}</math> of points with rational coordinates is dense in <math>\Gamma</math>; hence every continuous function <math>f:\Gamma \to \mathbb R</math> is determined by its restriction to <math>S</math>, so there can be at most <math>|\mathbb R|^{|S|} = 2^{\aleph_0}</math> many continuous real-valued functions on <math>\Gamma</math>. # On the other hand, the real line <math>L:=\{(p,0): p\in \mathbb R\}</math> is a closed discrete subspace of <math>\Gamma</math> with <math> 2^{\aleph_0}</math> many points. So there are <math>2^{2^{\aleph_0}} > 2^{\aleph_0}</math> many continuous functions from ''L'' to <math>\mathbb R</math>. Not all these functions can be extended to continuous functions on <math>\Gamma</math>. # Hence <math>\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== *Hedgehog space
==References== * Stephen Willard. ''General Topology'', (1970) Addison-Wesley {{ISBN|0-201-08707-3}}. * {{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=Counterexamples in Topology | orig-date=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover reprint of 1978 | isbn=978-0-486-68735-3 |mr=507446 | year=1995}} ''(Example 82)'' * {{planetmathref|urlname=NiemytzkiPlane|title= Niemytzki plane}}
Category:Topological spaces