[[Homotopy of maps]]
# Null-homotopic map

A morphism of topological spaces $f : X \to Y$ is said to be **null-homotopic** iff it is homotopic to a [[Constant map]]. #m/def/homotopy 
The same is said for loops under [[Homotopy of paths]].

## Properties

- In a [[Path connectedness|path-connected]] space null-homotopic maps form a single [[Homotopy of maps|homotopy class]] denoted $0$, since $0 \circ [f] = [f] \circ 0 = 0$.
- A space $X$ is [[Contractible space|contractible]] iff the identity $\id_{X}$ is null-homotopic.

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#state/tidy | #lang/en | #SemBr