On p2 of Algebraic Topology, Hatcher defines mapping cylinders as follows:

For a map $f: X \to\ Y$, the mapping cylinder $M_f$ is the quotient space of the disjoint union $(X \times I) \cup Y$ obtained by identifying each $(x,1) \in X \times I$ with $f(x) \in Y$.

Then, he makes the following remark:

Not all deformation retractions arise in this simple way from mapping cylinders. For example, the thick $\bf{X}$ deformation retracts to the thin $X$, which in turn deformation retracts to the point of intersection of its two crossbars. The net result is a deformation retraction of $\bf{X}$ onto a point, during which certain pairs of points follow paths that merge before reaching their final destination.

I'm unclear as to why the deformation retraction described in the paragraph above does not arise from a mapping cylinder. Hatcher seems to hint that the merging of paths is the reason why, but I do not see how merging of paths is inconsistent with the definition of a mapping cylinder.

Any thoughts?


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