Moore plane

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Topological space

title: "Moore plane" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["topological-spaces"] description: "Topological space" topic_path: "philosophy" source: "https://en.wikipedia.org/wiki/Moore_plane" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary 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

::figure[src="https://upload.wikimedia.org/wikipedia/commons/c/c2/Niemytzki_disk.png" caption="Open neighborhood of the Niemytzki plane, tangent to the x-axis"] ::

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:

  • Elements of the local basis at points (x,y) with y0 are the open discs in the plane which are small enough to lie within \Gamma.
  • 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 0}, & \mbox{if } q 0; \ { V_{\epsilon}(p):= {(p,0)} \cup {(x,y): (x-p)^2+(y-\epsilon)^2 0}, & \mbox{if } q = 0. \end{cases}

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.

::figure[src="https://upload.wikimedia.org/wikipedia/commons/2/2b/Moore_plane,Nemytskii's_tangent_disk_topology-_topological_space_representation_by_open_Neighbourhood.jpg" caption="Moore Plane graphic representation"] ::

Properties

Proof that the Moore plane is not normal

The fact that this space \Gamma 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: q0} of points with rational coordinates is dense in \Gamma; hence every continuous function f:\Gamma \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 \Gamma.
  2. On the other hand, the real line L:={(p,0): p\in \mathbb R} is a closed discrete subspace of \Gamma 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 \Gamma.
  3. Hence \Gamma 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.

References

  • Stephen Willard. General Topology, (1970) Addison-Wesley .
  • (Example 82)

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::

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