Double comb space
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 obtained by taking the standard comb space, and attaching another comb space to it at , where the new comb space is obtained by a half turn of the previous one about the point .
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.