---
tag: V25YYE26M1
kind: definition
name: Mapping cylinder
msc: [55P05, 54B17]
defines: EQVVDS0ZQT
uses: [0NQ1H2X495, ZXJKSB30WF, 0GBG8B49AA, 0VGFHTM75Y]
sources:
  - url: https://en.wikipedia.org/wiki/Mapping_cylinder
---

## Definition

Let $X$ and $Y$ be topological spaces, let $I = [0, 1]$, and let $f \colon X \to Y$ be a [[continuous map|0VGFHTM75Y]].
The *mapping cylinder* of $f$ is the space
$$M_f = \bigl( (X \times I) \sqcup Y \bigr) / \sim,$$
where $X \times I$ has the [[product topology|0GBG8B49AA]] and $\sqcup$ is the [[disjoint union|ZXJKSB30WF]].
The relation $\sim$ is the smallest equivalence relation with $(x, 1) \sim f(x)$ for all $x \in X$.
The space $M_f$ carries the [[quotient topology|0NQ1H2X495]].

## 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|CEK4SC20WK]] 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.