Contractible space: Difference between revisions

Revision as of 03:38, 9 November 2010

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

View other homotopy-invariant properties of topological spaces OR view all properties of topological spaces

Definition

Symbol-free definition

A topological space is said to be contractible if it satisfies the following equivalent conditions:

It is in the same homotopy class as a point
The identity map from the space to itself, is homotopic to a constant map, from the space to a particular point in that space
There is a single point which is a homotopy retract
It has a contracting homotopy

Definition with symbols

A topological space $X$ is said to be contractible if it has a contracting homotopy, viz a continuous map $f : X \times [0, 1] \to X$ and a point $x_{0} \in X$ such that $f (x, 0) = x$ and $f (x, 1) = x_{0}$ for all $x$ .

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	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

Metaproperties

Products

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

A direct product of contractible spaces is contractible. For full proof, refer: Contractibility is product-closed

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 of a contractible space is contractible. For full proof, refer: Contractibility is retract-hereditary

Subset-related properties

Closure

The closure of a contractible subset need not be contractible; in fact it need not even be path-connected. An example is the topologist's sine curve, which is contractible as it is the image of a path; however, it's closure is not even path-connected.

Interior

The interior of a contractible subset need not be contractible; in fact, it need not even be connected. An example is a wedge of two discs, whose interior is the disjoint union of their interiors.

Intersection

An intersection of contractible subsets need not be contractible. There is, however, a relation between the homology of the intersection and the homology of the union, when the contractible subsets are open (or more generally, when they are strong deformation retracts of neighbourhoods).

An example is the intersection of two semicircular paths on the circle, which is a pair of points.

Connected union

A connected union of contractible subsets need not be contractible.

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 PDF}^{More info}, Page 4 (formal definition)
Algebraic Topology by Edwin H. Spanier^{More info}, Page 25 (definition in paragraph)

@@ Line 39: / Line 39: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Stronger than::weakly contractible space]] || [[path-connected space]], all [[homotopy group]]s vanish || || || {{intermediate notions short|weakly contractible space|contractible space}}
+| [[Stronger than::weakly contractible space]] || [[path-connected space]], all [[homotopy group]]s vanish || [[contractible implies weakly contractible]] || [[weakly contractible not implies contractible]] || {{intermediate notions short|weakly contractible space|contractible space}}
 |-
-| [[Stronger than::multiply connected space]] || [[path-connected space]], first <math>k</math> [[homotopy group]]s vanish for <math>k \ge 2</math> || || || {{intermediate notions short|multiply connected space|contractible space}}
+| [[Stronger than::multiply connected space]] || [[path-connected space]], first <math>k</math> [[homotopy group]]s vanish for <math>k \ge 2</math> || || the <math>n</math>-sphere <math>S^n</math> is <math>(n-1)</math>-connected but not <math>n</math>-connected. || {{intermediate notions short|multiply connected space|contractible space}}
 |-
 | [[Stronger than::simply connected space]] || [[path-connected space]], [[fundamental group]] is [[trivial group|trivial]] || || || {{intermediate notions short|simply connected space|contractible space}}