Double comb space

Definition
The double comb space is a subset of $$\R^2$$ obtained by taking the standard defining ingredient::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 $$(0,1)$$.



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.

Related counterexamples

 * Closed topologist's sine curve