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I have to find in terms of standard spaces (that is interval, circle, disk, sphere, cone, etc.) the reduced cone (i.e. cone in category of pointed spaces) $\operatorname{Cone}(*)$ where $\{*\}$ is one point set.

We aren't discussing any topology.

Sorry for my English, it's not my native language.

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Your English is more or less fine (if I may say so, not being a native speaker myself), but there are three mostly language-independent things you should work on: firstly, you should accept some answers to your questions; secondly, spacing: generally, you put a space after each comma "," , and never before, and you put a space before (not after) the opening parenthesis "(", and after a closing parenthesis ")", and you should also use line breaks sometimes to make what you write readable, thirdly, you should put mathematical symbols and names in dollar symbols so as to TeXify them. – tomasz Oct 21 '12 at 18:56

The reduced cone over a pointed space is (as I recall) the cone over the space with the fiber over the base point collapsed. Precisely, if $(X,x_0)$ is a topological space, its reduced cone is defined by $$(X\times I)/(X\times\{0\}\cup \{x_0\}\times I).$$ At this stage, it will probably be easier for you to draw a picture for yourself and figure out the answer. I'll leave you with a tip: In practice, the way I think about the reduced cone is by first taking the cone, then doing an additional collapse along the fiber over the base point. So take the cone of $\{*\}$ and then collapse out the fiber over the base point of that space.

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Thank you for your help, but I can't use it because I defined Cone(X) as union of all rails between points from X and one external point, the top of Cone. – dandan Oct 20 '12 at 21:12
Sorry, what's a "rail"? When you said the rail between $\{*\}$ and the cone point, I thought you meant just the interval. What topology are you using on this union? Here- would you mind just editing your precise definition into the question? – Neal Oct 20 '12 at 21:59
Yes, by rail I meant interval. I change the form of question, that's the question I got. We didn't mention topology. – dandan Oct 21 '12 at 18:38

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