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
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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 $\mathbb {R} ^{2}$.

| class="sortable" border="1" ! Property !! Satisfied? !! Explanation !! Corollary properties satisfied/dissatisfied |- | closed sub-Euclidean space || Yes || || satisfies: sub-Euclidean space, completely metrizable space |- | locally Euclidean space || No || Not Euclidean around point of intersection || dissatisfies: manifold |- ! colspan="4" style="text-align:center; background: white"| 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 || || |- ! colspan="4" style="text-align:center; background: white"| 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 |- ! colspan="4" style="text-align:center; background: white"| Compactness |- | locally compact space || Yes || || |- | compact space || No || || |- | paracompact space || Yes || via metrizability || |- ! colspan="4" style="text-align:center; background: white"| Countability |- | second-countable space || Yes || || satisfies: first-countable space, separable space |- ! colspan="4" style="text-align:center; background: white"| Miscellaneous |- | homogeneous space || No || point of intersection is not similar to other points || |}