Contractible space

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.

Extreme and basic examples

 * The one-point space is contractible.
 * Any Euclidean space is contractible.
 * The closed unit disk in any dimension is contractible.
 * Compact manifolds in dimension one or more, such as the circle, are not contractible.

Intuition behind examples
Contractibility is, fundamentally, a global property of topological spaces. It is not possible to examine a small part of the space and conclude that it is contractible, nor does examining a small part of the space allow us to rule out the possibiilty that it is contractible. For the intuition behind the former, note that we can attach non-contractible pieces (like circles) far off from the part of the space we are looking at. For the intuition behind the latter claim, note that we can embed any topological space as a closed subspace of a contractible space, namely, its cone space.

For this reason, when looking for examples or counterexamples, we need to focus on the global structure.

Examples from topological construction
One thing to keep in mind is that since the definition of contractibility invokes the closed unit interval, it is likely that any effort to construct contractible spaces will invariably involve dealing with the real numbers. The most topologically general way of constructing a contractible space is as the cone space of an arbitrary topological space. One way of thinking of this cone space is as a literal cone that fills in between the space and a point. Up until the very tip of the cone, the cross-sections look homeomorphic to the topological space.

Examples from geometry
A topologically star-like space is a classic example of a contractible space. A topological space is termed topologically star-like if it is homeomorphic to a star-like subset of Euclidean space. A star-like subset of Euclidean space is a subset for which there exists a point in it such that for every other point in it, the line segment joining the points is completely inside the space.

A topologically star-like space is contractible, and can in fact be contracted to any point relative to which it is a star through a straight-line homotopy, i.e., moving each point toward the center in a straight line. The contracting homotopy fixes the center, and therefore, the space is in fact a SDR-contractible space.

Note that, if also compact, a topologically star-like space is homeomorphic to the cone space of its boundary. Otherwise, the space is still almost a cone space: it is a subspace of the cone space that contains the full complement of the base and an arbitrary subset of the base. Nonetheless, it is important to note that the condition of being star-like also carries various geometric implications (in particular, from being a sub-Euclidean space) that are not satisfied for arbitrary cone spaces.

A topologically convex space is a (non-empty) space that is homeomorphic to a convex subset of Euclidean space. Any topologically convex space is topologically star-like, and any point can be taken as the center. An example of a topologically star-like space that is not topologically convex is a pair of intersecting lines.

It is possible to construct spaces that are not topologically star-like, but still contractible. For instance, any geometric realization of a tree is contractible, but if the tree has more than one point with degree greater than two, it is not topologically star-like. As a related example, a set of parallel lines combined with one line that intersects all of them form a contractible space that is not topologically star-like.

Incomparable properties
Contractibility is incomparable with most of the interesting separation and compactness properties.

Broad argument for why contractibility cannot imply any meaningful separation or compactness property: The cone space over any topological space is contractible. In particular, since any topological space arises as a closed subspace of its cone space (namely, the "base" of the space), every topological space arises as a closed subspace of a contractible space. Therefore, contractible cannot imply any nontrivial property that is subspace-hereditary or even weakly hereditary (inherited by closed subsets).

Broad argument for why meaningful separation or compactness properties cannot imply contractibility: Most meaningful separation and compactness properties are satisfied by all compact manifolds. However, compact manifolds of dimension greater than one are not contractible. The simplest counterexample is generally the circle.

Some incomparable properties:


 * T0 space
 * T1 space
 * Hausdorff space
 * regular space
 * normal space
 * metrizable space
 * paracompact space
 * compact space

The property of being a contractible space is also incomparable with the property of being a locally contractible space. A contractible space need not be locally contractible. In fact, it need not even be locally connected! An example of a contractible space that is not locally contractible is the comb space. An example of a space that is locally contractible but not contractible is the circle (or, more generally, any compact manifold).

Conjunction with other properties

 * Contractible manifold: Contractible as well as a manifold
 * Contractible polyhedron: Contractible as well as a polyhedron, i.e., the geometric realization of a simplicial complex

Textbook references

 * , Page 330, Exercise 3 (definition introduced in exercise)
 * , Page 51 (formal definition)
 * , Page 18 (formal definition)
 * , Page 4 (formal definition)
 * , Page 25 (definition in paragraph)