Double comb space: Difference between revisions
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==Definition== | ==Definition== | ||
The '''double comb space''' is a subset of <math>\R^2</math> obtained by taking the standard [[comb space]], and attaching another comb space to it at <math>(0,1)</math>, where the new comb space is obtained by | The '''double comb space''' is a subset of <math>\R^2</math> obtained by taking the standard [[defining ingredient::comb space]], and attaching another comb space to it at <math>(0,1)</math>, where the new comb space is obtained by a half turn of the previous one about the point <math>(0,1)</math>. | ||
[[File:Doublecombspace.png|500px]] | |||
==Facts== | ==Facts== | ||
The double comb space is not [[contractible space|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]]. | The double comb space is not [[contractible space|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-space]]s; in fact it fails even for compact subsets of Euclidean space. | |||
==Related counterexamples== | |||
* [[Closed topologist's sine curve]] | |||
Latest revision as of 00:01, 21 December 2010
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.