Nemytskii 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 (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 = \$, then a topology may be defined on $\Gamma$ by taking a local basis $\mathcal(p,q)$ 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 $\\cup A$ 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
:$\mathcal(p,q) = \begin \, & \mbox q > 0; \\ \, & \mbox q = 0. \end$

Thus the subspace topology inherited by $\Gamma\backslash \$ 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 $\$ 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):
# On the one hand, the countable set $S:=\$ 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 $, \mathbb R, ^ = 2^$ many continuous real-valued functions on $\Gamma$.
# On the other hand, the real line $L:=\$ is a closed discrete subspace of $\Gamma$ with $2^$ many points. So there are $2^ > 2^$ many continuous functions from ''L'' to $\mathbb R$. Not all these functions can be extended to continuous functions on $\Gamma$.
# 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)''
* {{planetmathref, urlname=NiemytzkiPlane, title= Niemytzki plane

Topological spaces