Contractible spaces

Overview and basic definition

Definition: Nullhomotopic function, contractible space

A continuous function $f:X\to Y$ is nullhomotopic if $f$ is homotopic to a constant map. We say the space $X$ is contractible if the identity ${\text{id}}_X:X\to X$, defined by ${\text{id}}_X(x)=x$ for all $x\in X$, is nullhomotopic.

Equivalently, we say $X$ is contractible if the unique map $X\to \{\ast \}$ to the one-point space is a homotopy equivalence.

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Properties of contractible maps

Theorem ( Topology HW 9.3): Properties of contractible maps

• (i) If $X$ is contractible, then $X$ is path-connected.
• (ii) If $X$ is contractible, then any two continuous maps $f,g:Y\to X$ are homotopic.
• (iii) If $X$ is contractible and $Y$ is path-connected, then any two continuous maps $f,g:X\to Y$ are homotopic.

wip Consequently, any two maps into Euclidean space are homotopic?

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Example

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Review

• Show that two constant maps are homotopic if and only if they lie in the same pa