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It is easy to show that for any topological space $X$, the cone $CX$ is contractible. I am interested in the converse. If $Y$ is a contractible space, is $Y$ homeomorphic to $CX$ for some topological space $X$?

I was told the answer is no. However, I haven't been able to find a counterexample. I have two questions:

  • Is there a nice non-constructive way to see that counter-examples "should"/must exist?
  • Does anyone have a counter-example?
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Is an open interval a cone? –  MartianInvader Jan 16 '13 at 0:52
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Any tree with more than one vertex works. –  user641 Jan 16 '13 at 3:29
    
In my first draft of the question, I had relaxed the condition to dense subspaces of cones, specifically to include open intervals and balls, etc. For some reason I forgot to include those when I posted it... –  espen180 Jan 16 '13 at 9:39
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3 Answers

up vote 5 down vote accepted

A whole family of counter examples comes from finite spaces. Clearly, no finite nonempty space can be the cone of anything. But contractible finite spaces exist (there is even an entire relevant book, "Algebraic topology of finite topological spaces and applications", Barmak).

One way to quickly convince yourself that the answer must be 'no' is that cones are strongly related to $[0,1]$ (by construction!). So, any cone will have to admit some $\mathbb R$-like properties. However, contractibility is a very refined topological property and it is highly unlikely that just because a space is contractible that it will be strongly related to the real numbers. Maybe this counts as a non-constructive way to see that counterexamples "should" exist.

For an explicit non-finite example, consider http://en.wikipedia.org/wiki/House_with_two_rooms

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Just an aside: in finite spaces there is the related notion of non-Hausdorff cone, to which the question could apply and again not every contractible finite space is a nh cone –  user17786 Jan 16 '13 at 8:52
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A fun counterexample: written in 1-dimensional strokes, the letter $\pi$.

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In Hatcher's algebraic topology the exercise of first chapter gives a example that there is a space which is contractible but not homotopic to a point.(i.e. the space can be contracted to one point but in the process you must move the point) And we all know that a cone can always be deformation retracted to a point(thanks for pointing out my fault). So I think that counterexample shows everything.

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I think that this answer is wrong. You are probably thinking about math.stackexchange.com/questions/150751/… but a contractible space does have the homotopy type of a point. –  user27126 Jan 16 '13 at 2:21
    
Yes this is one of the examples. I didn't said that a contractible space is homotopic to a point but a cone... –  lee Jan 16 '13 at 2:36
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But every contractible space, including the one you mention, is homotopic to a point! I think what you mean to say is there's no strong deformation retraction of the space to a point (ie a deformation retraction that holds the point fixed), which is always true of a cone. –  MartianInvader Jan 16 '13 at 16:00
    
@MartianInvader Yes, I'm sorry for my mistake. –  lee Jan 17 '13 at 11:07
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