# Pair of intersecting lines

From Topospaces

This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces

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

A **pair of intersecting lines** is the underlying topological space of any subset of Euclidean space (of dimension two or higher) that comprises two distinct and intersecting lines. An example is the set:

This is the pair of the -axis and the -axis in .

## Properties

Property | Satisfied? | Explanation | Corollary properties satisfied/dissatisfied |
---|---|---|---|

Similarity to Euclidean space | |||

closed sub-Euclidean space | Yes | satisfies: sub-Euclidean space, completely metrizable space | |

locally Euclidean space | No | Not Euclidean around point of intersection | dissatisfies: manifold |

Separation and metrizability | |||

metrizable space | Yes | satisfies: binormal space, normal space, perfectly normal space, completely normal space, regular space, Hausdorff space, paracompact Hausdorff space, paracompact space | |

CW-space | Yes | ||

polyhedron | Yes | ||

Connectedness | |||

SDR-contractible space | Yes | satisfies: contractible space, simply connected space, path-connected space | |

locally contractible space | Yes | satisfies: locally path-connected space, locally simply connected space, semilocally simply connected space, locally connected space | |

Compactness | |||

locally compact space | Yes | ||

compact space | No | ||

paracompact space | Yes | via metrizability | |

Countability | |||

second-countable space | Yes | satisfies: first-countable space, separable space | |

Uniformness and self-similarity | |||

homogeneous space | No | point of intersection is not similar to other points | |

uniformly based space | No | basis sets containing the point of intersection look different from basis sets far away | dissatisfies: self-based space |

Topological equivalence to convex spaces | |||

topologically convex space | No | not convex around point of intersection: the rays emerging from it must be put in pairs, and convexity is violated for points on two unpaired rays | |

topologically star-like space | Yes | star-like relative to the point of intersection | satisfies: SDR-contractible space, contractible space |