Pair of intersecting lines

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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:

\{ (x,y) \in \R^2 \mid xy = 0 \}

This is the pair of the x-axis and the y-axis in \R^2.

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