Mapping cylinder

Remarks

The map $j \colon X \to M_f$, $x \mapsto [(x, 0)]$, is an embedding, and $Y$ embeds in $M_f$ as well. The map $r \colon M_f \to Y$ with $r[(x, t)] = f(x)$ and $r(y) = y$ is a homotopy equivalence, $Y$ is a strong deformation retract of $M_f$, and $f = r \circ j$. Example: for $Y$ a point, $M_f$ is the cone $CX = (X \times I) / (X \times \set{1})$. Example: for $f = \id_X$, $M_f$ is homeomorphic to $X \times I$. Non-example: the space $(X \times I) \sqcup Y$ without the identification is not $M_f$; for $X$ and $Y$ nonempty it is disconnected even when $X$ and $Y$ are connected, while $M_f$ is then connected.