# Definition:Homotopy/Null-Homotopic

< Definition:Homotopy(Redirected from Definition:Null-Homotopic)

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## Definition

Let $X$ and $Y$ be topological spaces.

Let $f: X \to Y$ be a continuous mapping.

Then:

- $f$ is
**null-homotopic**

- there exists a constant mapping $g: X \to Y$ such that $f$ and $g$ are homotopic.

### Null-Homotopy

Let $y \in Y$.

Let $g: X \to Y$ be a constant mapping, where the image of $g$ is equal to $\set {y}$.

Let $H : X \times \closedint 0 1 \to Y$ be a continuous mapping such that:

- $\forall x \in X : \map H {x, 0} = \map f x $

- $\forall x \in X : \map H {x, 1} = \map g x = y $

Then $H$ is called a **null-homotopy**.

## Also known as

Some texts spell this as **nulhomotopic**, or **nullhomotopic**.

## Sources

- 2000: James R. Munkres:
*Topology*(2nd ed.): $\S 51$ - 2011: John M. Lee:
*Introduction to Topological Manifolds*(2nd ed.) ... (previous) ... (next): $\S 7$: Homotopy and the Fundamental Group. Homotopy