Is the continuous map between CW-complexes a cofibration?

Question

If $f:A \rightarrow X$ is a continuous map between CW-complexes, then is $f$ necessarily a cofibration? I know that when $A$ is a subcomplex of $X$ and $f$ is the inclusion, the conclusion is true. Also, when $f$ is cellular, we can use the mapping cylinder to show the conclusion is true. But how about the general map $f$?

Answer 1

A cofibration is a closed inclusion (injective with closed image) for Hausdorff spaces and so not all continuous maps between CW-complexes are cofibrations. Example $f\colon\{1,2\}\rightarrow\{1\}$.