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


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).



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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