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It is to show, that for a well-pointed space $X$ its pathspace $PX$ is well-pointed.

$X$ being well pointed means that ${x} \to X$ is a cofibration. Maybe someone can give me a little hint, I don't see how to use the cofibration diagram of $X$ for the cofibration diagram of $PX$.

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Hint: Show that X-->PX is a cofibration. Show that a composition of cofibrations is a cofibration. –  Dylan Wilson Nov 9 '12 at 21:51
    
I solved it now by a lemma giving me equivalence between cofibration <-> local contraction with function from X -> I. I think we have PX with basepoint here, so only paths starting at same point x. So are you sure, that X -> PX even is a cofibration? I guess you were thinking of PX containing loops starting anywhere, so X is sent on the constant paths. I doubt this is a cofibration. –  x x Nov 12 '12 at 19:05
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