Difference between revisions of "Contractible space"
From Encyclopedia of Mathematics
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| − | A space is | + | {{TEX|semi-auto}}{{TEX|done}} |
| + | A [[Topological space|topological space]] $X$ that is homotopy equivalent (see [[Homotopy type|Homotopy type]]) to a one-point space; i.e., if there is a point $x \in X$ and a [[Homotopy|homotopy]] from $\operatorname{id}: X \rightarrow X$ to the unique mapping $p : X \rightarrow \{ x \}$. Such a mapping is called a contraction. | ||
| − | A | + | The [[Cone|cone]] over $X$ is contractible. For a [[Pointed space|pointed space]] $( X , * )$, the requirement for contractibility is that there is a base-point-preserving homotopy from $\operatorname{id}: ( X , * ) \rightarrow ( X , * )$ to the unique mapping $p : ( X , * ) \rightarrow ( * , * )$. |
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| + | A space is contractible if and only if it is a [[Retract|retract]] of the [[Mapping cylinder|mapping cylinder]] of any constant mappping $p : X \rightarrow \{ x \}$. | ||
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| + | A set $X \subset {\bf R} ^ { n }$ is starlike with respect to $x _ { 0 } \in X$ if for any $x \in X$ the segment $[ x _ { 0 } , x ]$ lies in $x$. Convex subsets and starlike subsets in ${\bf R} ^ { n }$ are contractible. | ||
====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> C.T.J. Dodson, P.E. Parker, "A user's guide to algebraic topology" , Kluwer Acad. Publ. (1997)</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)</td></tr></table> |
Latest revision as of 16:58, 1 July 2020
A topological space $X$ that is homotopy equivalent (see Homotopy type) to a one-point space; i.e., if there is a point $x \in X$ and a homotopy from $\operatorname{id}: X \rightarrow X$ to the unique mapping $p : X \rightarrow \{ x \}$. Such a mapping is called a contraction.
The cone over $X$ is contractible. For a pointed space $( X , * )$, the requirement for contractibility is that there is a base-point-preserving homotopy from $\operatorname{id}: ( X , * ) \rightarrow ( X , * )$ to the unique mapping $p : ( X , * ) \rightarrow ( * , * )$.
A space is contractible if and only if it is a retract of the mapping cylinder of any constant mappping $p : X \rightarrow \{ x \}$.
A set $X \subset {\bf R} ^ { n }$ is starlike with respect to $x _ { 0 } \in X$ if for any $x \in X$ the segment $[ x _ { 0 } , x ]$ lies in $x$. Convex subsets and starlike subsets in ${\bf R} ^ { n }$ are contractible.
References
| [a1] | C.T.J. Dodson, P.E. Parker, "A user's guide to algebraic topology" , Kluwer Acad. Publ. (1997) |
| [a2] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) |
How to Cite This Entry:
Contractible space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contractible_space&oldid=16066
Contractible space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contractible_space&oldid=16066
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article