{{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, 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 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== <references>
<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>
</references>
Category:Fixed points (mathematics) Category:Topology Category:Topological spaces
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