This may be a weird question but I really want to know the answer:
Hatcher p.2:
Why is the name 'cylinder' used in this instance? I don't think that this quotient space, namely the mapping cylinder, is homeomorphic to a cylinder. At the beginning of the chapter, he says that "… it should be read in this informal spirit, skipping bits here and there." So should I just ignore this?
Best Answer
First, one constructs a cylinder over the space $X$, that is, $X\times I$ (here $I=[0,1]$). Then one maps the top of the cylinder into $Y$. You can think on it as gluing the top $X\times \{1\}$ with the image $f(X)$.