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:

$$\! (x,1) \sim f(x)$$

Here, $$I = [0,1]$$ is the unit interval.

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.