# Mapping-cone construction

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The construction that associates with every continuous mapping $f : X \rightarrow Y$ of topological spaces the topological space $C _ {f} \supset Y$ obtained from the topological sum (disjoint union) $C X \oplus Y$( here $C X = ( X \times [ 0 , 1 ] ) / ( X \times \{ 0 \} )$ is the cone over $X$) by identifying $x \times \{ 1 \} = f ( x)$, $x \in X$. The space $C _ {f}$ is called the mapping cone of $f$. If $X$ and $Y$ are pointed spaces with distinguished points $x \in X$, $y \in Y$, then the generator $x \times [ 0 , 1 ]$ of $C X$ is contracted to a point, and $C _ {f}$ is said to be the collapsed mapping cone of $f$. For an arbitrary pointed topological space $K$, the sequence $X \rightarrow ^ {f} Y \subset C _ {f}$ induces an exact sequence
$$[ X , K ] \leftarrow [ Y , K ] \leftarrow [ C _ {f} , K ]$$
of pointed sets. The mapping $f$ is homotopic to the constant mapping to the distinguished point if and only if $Y$ is a retract of $C _ {f}$( cf. Retract of a topological space).