Fixed-point property

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. But another use is in order theory, where a partially ordered set "P" is said to have the fixed point property if every increasing function on "P" has a fixed point.
Definition.
Let "A" be an object in the concrete category C. Then "A" has the "fixed-point property" if every morphism (i.e., every function) formula_1 has a fixed point.
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 formula_2 has a fixed point.
Examples.
Singletons.
In the category of sets, the objects with the fixed-point property are precisely the singletons.
The closed interval.
The closed interval [0,1] has the fixed point property: Let "f": [0,1] → [0,1] be a continuous mapping. If "f"(0) = 0 or "f"(1) = 1, then our mapping has a fixed point at 0 or 1. If not, then "f"(0) > 0 and "f"(1) − 1 < 0. Thus the function "g"("x") = "f"("x") − x is a continuous real valued function which is positive at "x" = 0 and negative at "x" = 1. By the intermediate value theorem, there is some point "x"0 with "g"("x"0) = 0, which is to say that "f"("x"0) − "x"0 = 0, and so "x"0 is a fixed point.
The open interval does "not" have the fixed-point property. The mapping "f"("x") = "x"2 has no fixed point on the interval (0,1).
The closed disc.
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.
Topology.
A retract "A" of a space "X" with the fixed-point property also has the fixed-point property. This is because if formula_3 is a retraction and formula_1 is any continuous function, then the composition formula_5 (where formula_6 is inclusion) has a fixed point. That is, there is formula_7 such that formula_8. Since formula_7 we have that formula_10 and therefore formula_11
A topological space has the fixed-point property if and only if its identity map is universal.
A product of spaces with the fixed-point property in general fails to have the fixed-point property even if one of the spaces is the closed real interval.
The FPP is a topological invariant, i.e. is preserved by any homeomorphism. The FPP is also preserved by any retraction.
According to 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. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize 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.