Double comb space

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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 double comb space is a subset of \R^2 obtained by taking the standard comb space, and attaching another comb space to it at (0,1), where the new comb space is obtained by a half turn of the previous one about the point (0,1).

Doublecombspace.png

Facts

The double comb space is not contractible, but all its homology, homotopy, and cohomology groups vanish. Specifically, there is a continuous bijective map to the double comb space from a 1-CW-space (the underlying graph of the double comb space) which is not a homeomorphism but induces an isomorphism of the associated singular chain complex. Further, the map is a weak homotopy equivalence.

The double comb space is thus an illustration of the fact that Whitehead's theorem fails to extend to spaces beyond CW-spaces; in fact it fails even for compact subsets of Euclidean space.

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