Mapping cylinder

Definition

Let $f:X\to Y$ be a function. Then the mapping cylinder of $f$ is defined as the quotient of the disjoint union of $X\times I$ with $Y$, modulo the equivalence relation:

$(x,1)\simeq f(x)$

Facts

The significance of the mapping cylinder is that it is homotopy-equivalent to $Y$, and moreover the inclusion of $X$ (say via $x\mapsto (x,0)$) in the mapping cylinder is equivalent to the map $f$.

Thus, starting from an arbitrary continuous map, we have got a homotopy-equivalent map which is an inclusion.

Further, the inclusion of $X$ in the mapping cylinder is a cofibration, which makes it even nicer.