Fixed-point space

The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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 $f:X\rightarrow X$ has a fixed point, a point $x$ for which $f(x)=x$ .^[1]

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.^[1]

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.^[1]

References

^ ^a ^b ^c Granas, Andrzej; Dugundji, James (2003), Fixed Point Theory, Springer Monographs in Mathematics, New York: Springer-Verlag, p. 2, doi:10.1007/978-0-387-21593-8, ISBN 0-387-00173-5, MR 1987179

This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.

[gd-1] Granas, Andrzej; Dugundji, James (2003), Fixed Point Theory, Springer Monographs in Mathematics, New York: Springer-Verlag, p. 2, doi:10.1007/978-0-387-21593-8, ISBN 0-387-00173-5, MR 1987179

[1]