Closed infinite broom

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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

The closed infinite broom is the subset of the Euclidean plane obtained as the union of the following line segments: the line segment joining (0,0) to (1,0), and the line segment joining (0,0) to (1,1/n) where n varies over the natural numbers.

Closedinfinitebroom.png