Weakly contractible space

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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 weakly contractible if it satisfies the following equivalent conditions. The empty space is generally excluded from consideration when considering the question of weak 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 \ge 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

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
contractible space admits a contracting homotopy (direct from definition) weakly contractible not implies contractible (however, Whitehead's theorem says that weakly contractible equals contractible when we restrict to CW-spaces) |FULL LIST, MORE INFO
topologically convex space homeomorphic to a convex subset of Euclidean space (via contractible) (via contractible) Contractible space|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
acyclic space all the homology groups are the same as those of a point weakly contractible implies acyclic |FULL LIST, MORE INFO
rationally acyclic space all the homology groups over the rationals are the same as those of a point Acyclic space|FULL LIST, MORE INFO
simply connected space the fundamental group is trivial |FULL LIST, MORE INFO
simple space the fundamental group is abelian and it acts trivially on all the higher homotopy groups Simply connected space|FULL LIST, MORE INFO
path-connected space any two points can be connected by a path |FULL LIST, MORE INFO

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.