# 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 |