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 has a fixed point, a point for which .[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
edit- ^ 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
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