A mathematical object X has the fixed-point property if every suitably well-behaved mapping from X to itself has a fixed point.
The term is most commonly used to describe topological spaces on which every continuous mapping has a fixed point.
Let A be an object in the concrete category C. Then A has the fixed-point property if every morphism (i.e., every function)
The most common usage is when C = Top is the category of topological spaces.
Then a topological space X has the fixed-point property if every continuous map
In the category of sets, the objects with the fixed-point property are precisely the singletons.
The closed interval [0,1] has the fixed point property: Let f: [0,1] → [0,1] be a continuous mapping.
The closed interval is a special case of the closed disc, which in any finite dimension has the fixed-point property by the Brouwer fixed-point theorem.
A topological space has the fixed-point property if and only if its identity map is universal.
According to the Brouwer fixed-point theorem, every compact and convex subset of a Euclidean space has the FPP.
More generally, according to the Schauder-Tychonoff fixed point theorem every compact and convex subset of a locally convex topological vector space has the FPP.
In 1932 Borsuk asked whether compactness together with contractibility could be a sufficient condition for the FPP to hold.
The problem was open for 20 years until the conjecture was disproved by Kinoshita who found an example of a compact contractible space without the FPP.