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I can't image this, Someone can give a clear illustration?

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This is covered in Chapter 0 (the introductory chapter) of Algebraic Topology by Allen Hatcher.

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    $\begingroup$ Yes, but is not clearly for me. $\endgroup$
    – gylns
    Mar 9, 2010 at 0:28
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    $\begingroup$ Hatcher gives a pretty lucid description. What part of it is not clear? He suggests visualizing a thickening of the space as made out of clay: have you tried using playdough? I have resorted to playdough many times when my visual imagination failed me. $\endgroup$
    – Steven Gubkin
    Mar 9, 2010 at 0:35
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    $\begingroup$ It sounds like this is a language issue. Imagine a drinking glass full of wax. It's a solid object. By melting the wax and draining the liquid wax, you in effect "hollow out the chamber" -- the chamber being the glass full of wax. The hollow chamber is the empty glass. $\endgroup$
    – Ryan Budney
    Mar 9, 2010 at 1:04
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    $\begingroup$ You can realize the deformation-retraction as a sequence (concatenation) of "elementary collapse" operations. In particular you can write the map as a piecewise construction, made of composites of rational polynomial functions. These elementary collapses appear in many places in Hatcher's book -- the main construction in Proposition 0.16 of Chapter 0 (page 15) is the first such explicit construction, I think. $\endgroup$
    – Ryan Budney
    Mar 10, 2010 at 5:43
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    $\begingroup$ You might want to take a look at Marshall Cohen's book "A Course in Simple Homotopy Theory". He's quite explicit about these sorts of details. $\endgroup$
    – Ryan Budney
    Mar 10, 2010 at 6:28

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