Important Facts Involving Cofibrations

This map is obviously continuous and satisfies all the required properties. 

Let us check that the maps h and u satisfy the conditions of Definition 7.4  

There are several statements that intuitively seem likely to be always true, which in fact require the extra condition that the maps be cofibrations. Here 

The intuitive statement that taking a cone is the same as taking a quotient can also be made precise as follows. 

Since we have a homeomorphism Cone(i)/C = X/X, which commutes with the quotient maps, and C is contractible inside of Cone(i), the corollary follows from Proposition 7.8.  

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