Weakly contractible space

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

This is a variation of contractible space. View other variations of contractible space

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.

As we see below, each of the definitions (implicitly or explicitly) implies that the space is a path-connected space.

No.	Shorthand	A topological space is termed weakly contractible if ...	A topological space $X$ is termed weakly contractible if ...
1	weakly homotopy-equivalent to a point	it is weakly homotopy-equivalent to a one-point space. Note that the definition in particular implies that the space is a path-connected space.	$X$ is path-connected, and it is in the same equivalence class as the one-point space under weak homotopy equivalence.
2	weakly homotopy-equivalent to a point (unique map to the point)	it is path-connected and the unique map from it to the one-point space is a weak homotopy equivalence of topological spaces (regardless of choice of basepoint).	(fix any point $x_{0}\in X$; the truth of this fact does not depend on the choice of $x_{0}$) $X$ is path-connected, and the unique map $f:X\to \{*\}$ to the one-point space (which is continuous by definition) induces isomorphisms $\pi _{f}:\pi _{n}(X,x_{0})\to \pi _{n}(\{*\},*)$ on all the homotopy groups.
3	weakly homotopy-equivalent to a point (arbitrary choice of basepoint)	it is path-connected and any map from a one-point space to it induces a weak homotopy equivalence of topological spaces	(fix any point $x_{0}\in X$; the truth of this fact does not depend on the choice of $x_{0}$) Suppose $g$ is the (continuous by definition) map from the one-point space $\{*\}$ to $X$ that sends the point $*$ to $x_{0}$. Then, $g$ induces isomorphisms $\pi _{g}:\pi _{n}(\{*\},*)\to \pi _{n}(X,x_{0}$ on all the homotopy groups.
4	all homotopy groups are trivial	it is path-connected (i.e., its set of path components has size one) and all its homotopy groups are trivial.	(fix any point $x_{0}\in X$; the truth of this fact does not depend on the choice of $x_{0}$) $X$ is path-connected and $\pi _{n}(X,x_{0})$ is the trivial group for all positive integers $n$.
5	any map from a sphere is nullhomotopic	it is path-connected and any continuous map from a sphere (of any finite dimension) to it is nullhomotopic.	$X$ is path-connected, and any continuous map $f:S^{n}\to X$ for any $n\geq 1$ is nullhomotopic, i.e., we can construct a homotopy from that map to a map that sends $S^{n}$ to a single point. Note that restricting ourselves to homotopies that fix a basepoint does not change the strength of the definition.
6	any map from a connected CW-space is nullhomotopic	it is path-connected and any continuous map from a connected CW-space (the underlying space of a CW-complex) to it is nullhomotopic.	$X$ is path-connected and for any connected CW-space $Y$ and any continuous map $f:Y\to X$, $f$ is nullhomotopic. Note that restricting ourselves to homotopies that fix a basepoint does not change the strength of the definition.
7	any map from a connected polyhedron is nullhomotopic	it is path-connected and any continuous map from a connected polyhedron (the geometric realization of a simplicial complex) to it is nullhomotopic.	$X$ is path-connected and for any connected polyhedron $Y$ and any continuous map $f:Y\to X$, $f$ is nullhomotopic. Note that restricting ourselves to homotopies that fix a basepoint does not change the strength of the definition.
8	any map from a connected manifold is nullhomotopic	it is path-connected and any continuous map from a connected manifold to it is nullhomotopic.	$X$ is path-connected and for any connected manifold $Y$ and any continuous map $f:Y\to X$, $f$ is nullhomotopic. Note that restricting ourselves to homotopies that fix a basepoint does not change the strength of the definition.
9	any map from a manifold is nullhomotopic	it is path-connected and any continuous map from a manifold to it is nullhomotopic.	$X$ is path-connected and for any manifold $Y$ and any continuous map $f:Y\to X$, $f$ is nullhomotopic. Note that restricting ourselves to homotopies that fix a basepoint does not change the strength of the definition.

Relation with other properties

Stronger properties

• Contractible space: The converse implication holds for CW-spaces, via Whitehead's theorem

Metaproperties

Products

This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products

Since the homotopy group of the product of two spaces is the product of their homotopy groups, the product of two weakly contractible spaces is again weakly contractible.

Retract-hereditariness

This property of topological spaces is hereditary on retracts, viz if a space has the property, so does any retract of it
View all retract-hereditary properties of topological spaces

Any retract, and more generally, any homotopically injective subspace of a weakly contractible space is again weakly contractible.