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 (also called Tychonoff space) which is not normal. It is named after Robert Lee Moore and Viktor Vladimirovich Nemytskii.

Definition

If \Gamma is the (closed) 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:

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

Proof that the Moore plane is not normal

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

References

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