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A [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120220/c1202201.png" /> that is homotopy equivalent (see [[Homotopy type|Homotopy type]]) to a one-point space; i.e., if there is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120220/c1202202.png" /> and a [[Homotopy|homotopy]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120220/c1202203.png" /> to the unique mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120220/c1202204.png" />. Such a mapping is called a contraction.
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The [[Cone|cone]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120220/c1202205.png" /> is contractible. For a [[Pointed space|pointed space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120220/c1202206.png" />, the requirement for contractibility is that there is a base-point-preserving homotopy from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120220/c1202207.png" /> to the unique mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120220/c1202208.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120220/c1202209.png" />.
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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 set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120220/c12022010.png" /> is starlike with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120220/c12022011.png" /> if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120220/c12022012.png" /> the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120220/c12022013.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120220/c12022014.png" />. Convex subsets and starlike subsets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120220/c12022015.png" /> are contractible.
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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><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>
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<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=50240
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article