In mathematics, a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

is said to be weakly contractible if all of its homotopy groups are trivial.

Property

It follows from

Whitehead's Theorem In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping ''f'' between CW complexes ''X'' and ''Y'' induces isomorphisms on all homotopy groups, then ''f'' is a homotopy equivalence. This result was ...

that if a

CW-complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This clas ...

is weakly contractible then it is

contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...

Example

Define

S^\infty

to be the

inductive limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any catego ...

of the spheres

S^n, n\ge 1

. Then this space is weakly contractible. Since

S^\infty

is moreover a CW-complex, it is also contractible. See

Contractibility of unit sphere in Hilbert space In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space ''H''. It states that the space GL(''H'') of invertible bounded endomorphisms of ''H'' is suc ...

for more. The

Long Line Long line or longline may refer to: *'' Long Line'', an album by Peter Wolf * Long line (topology), or Alexandroff line, a topological space *Long line (telecommunications), a transmission line in a long-distance communications network *Longline fi ...

is an example of a space which is weakly contractible, but not contractible. This does not contradict Whitehead theorem since the Long Line does not have the homotopy type of a CW-complex. Another prominent example for this phenomenon is the Warsaw circle.

References

* Topology Homotopy theory {{Topology-stub