Editing Fixed-point space

{{Short description|Space where all functions have fixed points}}
In [[mathematics]], a [[Hausdorff space]] ''X'' is called a '''fixed-point space''' if it obeys a [[fixed-point theorem]], according to which every [[continuous function]]  <math>f:X\rightarrow X</math> has a [[fixed point (mathematics)|fixed point]], a point <math>x</math> for which <math>f(x)=x</math>.{{r|gd}}

For example, the [[closed unit interval]] is a fixed point space, as can be proved from the [[intermediate value theorem]]. The [[real line]] is not a fixed-point space, because the continuous function that adds one to its argument does not have a fixed point. Generalizing the unit interval, by the [[Brouwer fixed-point theorem]], every [[compact set|compact]] bounded [[convex set]] in a [[Euclidean space]] is a fixed-point space.{{r|gd}}

The definition of a fixed-point space can also be extended from continuous functions of topological spaces to other classes of maps on other types of space.{{r|gd}}

==References==
{{reflist|refs=

<ref name=gd>{{citation
 | last1 = Granas | first1 = Andrzej
 | last2 = Dugundji | first2 = James
 | doi = 10.1007/978-0-387-21593-8
 | isbn = 0-387-00173-5
 | mr = 1987179
 | page = [https://books.google.com/books?id=apLzBwAAQBAJ&pg=PA2 2]
 | publisher = Springer-Verlag | location = New York
 | series = Springer Monographs in Mathematics
 | title = Fixed Point Theory
 | year = 2003}}</ref>

}}

[[Category:Fixed points (mathematics)]]
[[Category:Topology]]
[[Category:Topological spaces]]


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