Contractible space - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

A topological space $X$ is contractible if there exist a point $x_0\in X$ and a homotopy

H:X\times[0,1]\to X,\qquad H(x,0)=x,\quad H(x,1)=x_0

for every $x\in X$ . Thus the identity map of $X$ is homotopic to a constant map. The homotopy is not required to keep $x_0$ fixed at intermediate times.

Examples and scope

Every nonempty convex subset of $\mathbb R^n$ is contractible: choose $x_0$ in it and use $H(x,t)=(1-t)x+tx_0$ . A one-point space is contractible. The empty space is not contractible under this convention, because no point $x_0$ exists.

Contractibility is a property of the space with its given topology; an ambient deformation that leaves the space is not a contraction of that space.