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This is the problem: Let X={a,b} be topology space with trivial topology and A={a} is one point set (obviously not open). Prove using definition of cofibration that inclusion A->X is cofibration.

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What have you tried? – Jason DeVito Oct 9 '12 at 12:53
I'm trying to construct extended homotopy F' but I get stuck because I get that the extended homotopy F' is equal to F on A, and on X\A it can be any continuos function – Lilika Oct 9 '12 at 13:10
Well, yes, and what's wrong with it? The simplest if you define the same for $b$ as what is given for $a$. – Berci Oct 9 '12 at 14:03
Isn't that too easy? Then I do nothing special, just define F'(x,t)=F(a,t) for every x in X and t in [0,1]. Shouldn't I prove something? – Lilika Oct 9 '12 at 14:07
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then you're done. I think, $b$ must be mapped the same as $a$ for continuity, but that's all. Go on to next exercise. – Berci Oct 9 '12 at 14:37
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