Moore plane explained

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

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 l{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 .

• 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

l{B}(p,q) = \begin{cases} \{ U _{\epsilon} (p,q): = \{(x,y): (x - p) ^{2+(y-q)} ^{2} < \epsilon ^{2} \} \mid \epsilon > 0 \}, & if q > 0 ; \ \{ V _{\epsilon} (p): = \{(p, 0 )\} \cup \{(x,y): (x - p) ^{2+(y-\epsilon)} ^{2} < \epsilon ^{2} \} \mid \epsilon > 0 \}, & 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.

Properties

• The Moore plane

\Gamma 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

\{(x, 0 )\in \Gamma | x\in R \} of \Gamma 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

\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 Q x Q: q> 0 \} of points with rational coordinates is dense in \Gamma ; hence every continuous function f:\Gamma \to R is determined by its restriction to S , so there can be at most | R| ^{|S|} = 2 ^{\aleph _{0}} many continuous real-valued functions on \Gamma .

1. On the other hand, the real line

L: = \{(p, 0 ): p\in 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 R . Not all these functions can be extended to continuous functions on \Gamma .

1. 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.

See also

• Hedgehog space

References

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