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Fixed-point space

Space where all functions have fixed points

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Source updated August 29, 2026

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 $f(x)=x$ .

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.

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

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This article is adapted from the Wikipedia article “Fixed-point space”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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