Comb space

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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

The comb space is defined as the following subset of \R^2 with the subspace topology: It is the union of [0,1] \times \{ 0 \}, \{ 0 \} \times [0,1], and all line segments of the form \{ 1/n \} \times [0,1] where n varies over the positive integers.

Combspace.png

Topological space properties

Properties it does satisfy

Properties it does not satisfy

Related spaces