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Moore plane

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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) that is not normal. It is named after Robert Lee Moore and Viktor Vladimirovich Nemytskii.

Contents

Definition

If Γ is the (closed) upper half-plane Γ = { ( x , y ) ∈ R 2 | y ≥ 0 } , then a topology may be defined on Γ by taking a local basis 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 Γ . Thus the subspace topology inherited by Γ ∖ { ( x , 0 ) | x ∈ 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 } ∪ 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

    B ( p , q ) = { { U ϵ ( p , q ) := { ( x , y ) : ( x − p ) 2 + ( y − q ) 2 < ϵ 2 } ∣ ϵ > 0 } , if  q > 0 ; { V ϵ ( p ) := { ( p , 0 ) } ∪ { ( x , y ) : ( x − p ) 2 + ( y − ϵ ) 2 < ϵ 2 } ∣ ϵ > 0 } , if  q = 0.

    Properties

  • The Moore plane Γ 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 ) ∈ Γ | x ∈ R } of Γ 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 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 ) ∈ Q × Q : q > 0 } of points with rational coordinates is dense in M; hence every continuous function f : M → R is determined by its restriction to S , so there can be at most | R | | S | = 2 ℵ 0 many continuous real-valued functions on M.
    2. On the other hand, the real line L := { ( p , 0 ) : p ∈ R } is a closed discrete subspace of M with 2 ℵ 0 many points. So there are 2 2 ℵ 0 > 2 ℵ 0 many continuous functions from L to 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.

    References

    Moore plane Wikipedia
    (Text) CC BY-SA