fixed-point property

The open interval does not have the fixed-point property.

A topological space has the fixed-point property if and only if its identity map is universal.

Then a topological space X has the fixed-point property if every continuous map has a fixed point.

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 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 retract A of a space X with the fixed-point property also has the fixed-point property.

Let A be an object in the concrete category C. Then A has the fixed-point property if every morphism (i.e., every function) has a fixed point.

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.