Moore plane
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
Properties it does satisfy
- Separable space
- First-countable space
- Completely regular space
- Locally normal space
- Countably metacompact space
Properties it does not satisfy
- Normal space: For full proof, refer: Moore plane is not normal
- Hereditarily separable space: The bounding line is an uncountable discrete subset
- Collectionwise Hausdorff space
- Metacompact space