Definition:Homotopy/Null-Homotopic

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.

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.

Some texts spell this as nulhomotopic, or nullhomotopic.