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| 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]]) which is not [[normal space|normal]]. It is named after [[Robert Lee Moore]] and [[Viktor Vladimirovich Nemytskii]].
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| ==Definition==
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| [[Image:Niemytzki disk.png|218x187|right|Open neighborhood of the Niemytzki plane, tangent to the x-axis]]
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| If <math>\Gamma</math> is the 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:
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| *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>. 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.
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| *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''.
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| That is, the local basis is given by
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| :<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>
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| ==Properties==
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| *The Moore plane <math>\Gamma</math> is [[Separable space|separable]], that is, it has a countable dense subset.
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| *The Moore plane is a [[Tychonoff space|completely regular Hausdorff space]] (i.e. [[Tychonoff space]]), which is not [[normal space|normal]].
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| *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.
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| *The Moore plane is [[first countable]], but not [[second countable]] or [[Lindelöf space|Lindelöf]].
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| *The Moore plane is not [[locally compact]].
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| *The Moore plane is [[countably metacompact]] but not [[metacompact]].
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| ==Proof that the Moore plane is not normal== | |
| The fact that this space ''M'' is not [[normal space|normal]] can be established by the following counting argument (which is very similar to the argument that the [[Sorgenfrey plane]] is not normal):
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| # 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 ''M''; hence every continuous function <math>f:M\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 ''M''.
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| # On the other hand, the real line <math>L:=\{(p,0): p\in \mathbb R\}</math> is a closed discrete subspace of ''M'' 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 ''M''.
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| # 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.
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| In fact, if ''X'' is a [[Separable space|separable]] topological space having an uncountable closed discrete subspace, ''X'' cannot be normal.
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| ==See also== | |
| *[[Moore space]]
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| *[[Hedgehog space]]
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| == References ==
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| * Stephen Willard. ''General Topology'', (1970) Addison-Wesley ISBN 0-201-08707-3.
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| * {{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]] | origyear=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | id={{MathSciNet|id=507446}} | year=1995}} ''(Example 82)''
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| * {{planetmathref|id=4247|title= Niemytzki plane}}
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| [[Category:Topological spaces]]
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