Pair of intersecting lines

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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:

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

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

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