Contractible space

This article defines a homotopy-invariant property of topological spaces, i.e. a property of homotopy classes of topological spaces

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Definition

Equivalent definitions in tabular format

A nonempty topological space is said to be contractible if it satisfies the following equivalent conditions. The empty space is generally excluded from consideration when considering the question of contractibility.

No.	Shorthand	A topological space is termed contractible if ...	A topological space $X$ is termed contractible if ...
1	homotopy-equivalent to a point	there is a homotopy equivalence of topological spaces between the topological space and a one-point space.	There exist continuous maps $f:X\to \{*\}$ and $g:\{*\}\to X$ such that $f\circ g$ is homotopic to the identity map on $\{*\}$ and $g\circ f$ is homotopic to the identity map on $X$. Here, $\{*\}$ is a one-point space. Also, note that the condition on $f\circ g$ is tautological, and the map $f$ is already uniquely determined, so all the action occurs in the definition of $g$ and the homotopy between $g\circ f$ and the identity map on $X$.
2	homotopy-equivalent to a point (arbitrary map)	any pair of maps between the space and a one-point space define a homotopy equivalence of topological spaces.	For any continuous map $f:X\to \{*\}$ and $g:\{*\}\to X$, $f\circ g$ is homotopic to the identity map on $\{*\}$ and $g\circ f$ is homotopic to the identity map on $X$. Here, $\{*\}$ is a one-point space. Also, note that the condition on $f\circ g$ is tautological, and the map $f$ is already uniquely determined, so all the action occurs in the definition of $g$ and the homotopy between $g\circ f$ and the identity map on $X$.
3	admits a contracting homotopy	there is a point in the space to which there is a contracting homotopy.	there exists a point $x_{0}\in X$ and a contracting homotopy that contracts $X$ to $x_{0}$. Explicitly, there exists a continuous map $F:X\times [0,1]\to X$ such that $F(x,0)=x$ for all $x$, and $F(x,1)=x_{0}$ for all $x$. Note that we do not assume or require that $F(x_{0},a)=x_{0}$ for all $a\in [0,1]$.
4	admits a contracting homotopy (arbitrary point)	the space admits a contracting homotopy to any point in it.	for any point $x_{1}\in X$, there is a contracting homotopy that contracts $X$ to $x_{1}$.
5	unique homotopy class of maps to it	for any other topological space, there is a unique homotopy class of maps from the other space to it.	for any topological space $Y$, and any two continuous maps $h_{1},h_{2}:Y\to X$, $h_{1}$ and $h_{2}$ are homotopic. In particular, any map from a topological space to it is nullhomotopic.
6	retract of cone space	it is a retract of its cone space.	the inclusion $\iota :X\to CX$ in the cone space $CX$ (as the base) has a one-sided inverse, $j:CX\to X$, i.e., $j$ is a continuous map such that $j\circ \iota$ is the identity map on $X$.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
product-closed property of topological spaces	Yes	contractibility is product-closed	If $X_{i},i\in I$ form a (finite or infinite) collection of contractible spaces, then the product of the $X_{i}$s, equipped with the product topology, is also contractible. In particular, if $X_{1},X_{2}$ are contractible, then $X_{1}\times X_{2}$ is also contractible.
retract-hereditary property of topological spaces	Yes	contractibility is retract-hereditary	If $A\subseteq X$ and $f:X\to A$ is a continuous retraction, and $X$ is contractible, then $A$ is contractible.
suspension-closed property of topological spaces	Yes	contractibility is suspension-closed	The suspension of a contractible space is contractible.
closure-preserved property of topological spaces	No	contractibility is not closure-preserved	It is possible to have a topological space $X$ and a subset $A$ of $X$ such that $A$ is contractible in the subspace topology, but the closure ${\overline {A}}$ is not.
interior-preserved property of topological spaces	No	contractibility is not interior-preserved	It is possible to have a topological space $X$ and a subset $A$ of $X$ such that $A$ is contractible in the subspace topology, but the interior of $A$ is not.
intersection-closed property of topological spaces	No	contractibility is not intersection-closed	It is possible to have a topological space $X$ and subsets $A,B$ of $X$ such that $A,B$ are both contractible in their respective subspace topology but $A\cap B$ is not.
connected union-closed property of topological spaces	No	contractibility is not connected union-closed	It is possible to have a topological space $X$ expressible as a union of subsets $A,B$, both contractible in their subspace topology, with $A\cap B$ nonempty, but $X$ itself not contractible.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
cone space over some topological space		cone space implies contractible
topologically star-like space				|FULL LIST, MORE INFO
topologically convex space	homeomorphic to a convex subset of Euclidean space	via star-like		Equiconnected space, Space in which every retraction is a deformation retraction, Topologically star-like space|FULL LIST, MORE INFO
suddenly contractible space	has a contracting homotopy that is also a sudden homotopy			|FULL LIST, MORE INFO
SDR-contractible space	has a contracting homotopy that is also a deformation retraction			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
weakly contractible space	path-connected space, all homotopy groups vanish	contractible implies weakly contractible	weakly contractible not implies contractible	|FULL LIST, MORE INFO
multiply connected space	path-connected space, first $k$ homotopy groups vanish for $k\geq 2$		the $n$-sphere $S^{n}$ is $(n-1)$-connected but not $n$-connected.	|FULL LIST, MORE INFO
simply connected space	path-connected space, fundamental group is trivial			Weakly contractible space|FULL LIST, MORE INFO
path-connected space	there is a path between any two points			Weakly contractible space|FULL LIST, MORE INFO
connected space	cannot be partitioned into disjoint nonempty subsets			|FULL LIST, MORE INFO
acyclic space	homology groups over $\mathbb {Z}$ all zero except zeroth homology group which is $\mathbb {Z}$			Weakly contractible space|FULL LIST, MORE INFO
rationally acyclic space	homology groups over $\mathbb {Q}$ all zero except zeroth homology group which is $\mathbb {Q}$			Acyclic space, Weakly contractible space|FULL LIST, MORE INFO
space with Euler characteristic one	Euler characteristic of the space is one.	(via acyclic)	Euler characteristic one not implies acyclic	Acyclic space|FULL LIST, MORE INFO

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 330, Exercise 3 (definition introduced in exercise)
• Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 51 (formal definition)
• An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by Joseph J. Rotman^{More info}, Page 18 (formal definition)
• Algebraic Topology by Allen Hatcher^{Full text PDFMore info}, Page 4 (formal definition)
• Algebraic Topology by Edwin H. Spanier^{More info}, Page 25 (definition in paragraph)