Contractible
المؤلف:
Hatcher, A.
المصدر:
Algebraic Topology. Cambridge, England: Cambridge University Press, 2002.
الجزء والصفحة:
...
8-5-2021
2042
Contractible
A set in 
which can be reduced to one of its points, say 
, by a continuous deformation, is said to be contractible. The transformation is such that each point of the set is driven to 
through a path with the properties that
1. Each path runs entirely inside the set.
2. Nearby points move on "neighboring" paths.
Condition (1) implies that a disconnected set, i.e., a set consisting of separate parts, cannot be contractible.
Condition (2) implies that the circumference of a circle is not contractible. The latter follows by considering two near points 
and 
lying on different sides of a point 
. The paths connecting 
and 
with 
are either opposite each other or have different lengths. A similar argument shows that, in general, for all 
, the 
-sphere (i.e., the boundary of the 
-dimensional ball) is not contractible.
A gap or a hole in a set can be an obstruction to contractibility. There are, however, examples of contractible sets with holes, for example, the "house with two rooms." In a case like this, it is not evident how to construct a transformation of the type described above. However, its existence is assured by the formal definition of contractibility of a set 
, namely that 
is homotopic to one of its points 
. This means that there is a continuous map ![F:[0,1]×X->X](https://mathworld.wolfram.com/images/equations/Contractible/Inline16.gif)
such that 
is the identity map and 
is the constant map sending each point to 
. Thus, 
describes a continuous path from 
to 
as 
varies from 0 to 1, and (1) is fulfilled. Moreover, since the map 
is also continuous with respect to the second component, the path starting at 
varies continuously with respect to 
, as required by (2).
REFERENCES:
Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, 2002.
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