Math 2.0

General topology Definition

Fixed-point property

Builds on Topological spaceContinuous functionIntermediate value theoremCircle

Definition

Let XX be a topological space. XX has the fixed-point property if for every continuous map f ⁣:X→Xf \colon X \to X there is a point x∈Xx \in X with f(x)=xf(x) = x.

Remarks

Example: the interval [0,1][0, 1] has the fixed-point property. For continuous f ⁣:[0,1]→[0,1]f \colon [0, 1] \to [0, 1], the function g(x)=f(x)−xg(x) = f(x) - x satisfies g(0)≥0g(0) \ge 0 and g(1)≤0g(1) \le 0, so gg has a zero by the intermediate value theorem. Non-example: the circle S1⊆ℂS^1 \subseteq \C fails it, because the rotation z↦−zz \mapsto -z is continuous and has no fixed point. If XX has the fixed-point property and YY is homeomorphic to XX, then YY has it as well. A retract of a space with the fixed-point property also has it. The Brouwer fixed-point theorem states that the closed unit ball of ℝn\R^n has the fixed-point property.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Circle, concept, no evidence levelCircleconceptContinuous function, concept, no evidence levelContinuous functionconceptLimit point of a set, definition, no evidence levelLimit point of a setdefinitionTopological space, concept, no evidence levelTopological spaceconceptLimit of a function, definition, no evidence levelLimit of a functiondefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionIntermediate value theorem, statement, no evidence levelIntermediate value th…statementFixed-point property, definition, no evidence levelFixed-point propertydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Circle concept
  2. Continuous function concept
  3. Limit point of a set definition
  4. Topological space concept
  5. Limit of a function definition
  6. Continuity at a point definition
  7. Continuous function definition
  8. Intermediate value theorem statement
  9. Fixed-point property definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Fixed-point_property
About this page
Tag93FKWZP2AZ
URIm2://93FKWZP2AZ
Kinddefinition
WritingW0 stub
MSC54H25, 55M20
DefinesFixed-point property
UsesTopological space, Continuous function, Intermediate value theorem, Circle
Changed2 November 2026

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