Moore plane
From Topospaces
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 |
