Moore plane

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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. It is named after Robert Lee Moore and Viktor Vladimirovich Nemytskii.

Definition[edit]

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

If \Gamma is the upper half-plane \Gamma = \{(x,y)\in\R^2 | y \geq 0 \}, then a topology may be defined on \Gamma by taking a local basis \mathcal{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 \Gamma. Thus the subspace topology inherited by \Gamma\backslash \{(x,0) | x \in \R\} is the same as the subspace topology inherited from the standard topology of the Euclidean plane.
  • 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

\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}

Properties[edit]

Proof that the Moore plane is not normal[edit]

The fact that this space M 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 Q\times \mathbb Q: q>0\} of points with rational coordinates is dense in M; hence every continuous function f:M\to \mathbb R is determined by its restriction to S, so there can be at most |\mathbb R|^ {|S|} = 2^{\aleph_0} many continuous real-valued functions on M.
  2. On the other hand, the real line L:=\{(p,0): p\in \mathbb R\} is a closed discrete subspace of M with  2^{\aleph_0} many points. So there are 2^{2^{\aleph_0}} > 2^{\aleph_0} many continuous functions from L to \mathbb R. Not all these functions can be extended to continuous functions on M.
  3. 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.

In fact, if X is a separable topological space having an uncountable closed discrete subspace, X cannot be normal.

See also[edit]

References[edit]