Moore plane

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This article describes a standard counterexample to some plausible but false implications. In other words, it lists a pathology that may be useful to keep in mind to avoid pitfalls in proofs
View other standard counterexamples in topology

Definition

The Moore plane or Niemitzky plane or tangent disk topology is defined as follows: as a set, it is the upper half-plane, along with the bounding real line. The topology is described by the following basis:

  • All open disks that lie completely inside the upper half-plane
  • For points which are on the bounding line, the union of such a point with an open disk tangent to the bounding line at that point

Topological space properties

Property Satisfied? Explanation Corollary properties satisfied/dissatisfied
Separation type
completely regular space Yes Moore plane is completely regular satisfies: regular space, Hausdorff space, T1 space, Urysohn space, Kolmogorov space
locally normal space Yes Moore plane is locally normal
normal space No Moore plane is not normal dissatisfies: metrizable space, CW-space, hereditarily normal space
collectionwise Hausdorff space No Moore plane is not collectionwise Hausdorff -- the bounding line is an uncountable discrete subset
Countability type
separable space Yes Any dense subset of the upper half plane in the usual topology is also dense in the Moore plane.
first-countable space Yes We can, at any point, take a local basis that only uses disks of rational radius (use tangent disks for points on the bounding line)
hereditarily separable space No the bounding line is an uncountable discrete subset
Compactness type
countably metacompact space Yes Moore plane is countably metacompact
metacompact space No Moore plane is not metacompact dissatisfies: paracompact space, compact space

External links

Definition links