# The mapping cylinder of CW complex

If $X,Y$ are CW complexes and $f$ a cellular map from $X$ to $Y$, is it true that $M_f$ is a CW complex?

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First, it is a general and well-known fact that the product of two CW complexes is a CW complex. So $X \times I$ is a CW complex. And, from the construction, $X = X \times \left\{0\right\}$ is a subcomplex.
Let $X$ be a CW complex and $A \subset X$ a subcomplex. Suppose $f: A \rightarrow Y$ is a cellular map (for $Y$ another CW complex). Then the space $X \cup_f Y$ is a CW complex.
Proof: We take as the cells the cells of $X$ not in $A$ and the cells of $Y$. This is kosher, because $X-A$ is the union of various cells, $A$ being a subcomplex. The reason we have to require $f: A \rightarrow Y$ to be cellular, though, is that the boundary of each $n$-cell has to be contained in a union of $n-1$-cells.