8. A space has "the fixed point property" if every continuous function f: X → Y leaves at least one fixed point, i.e. there exists such that x in X such that f(x)=x. Prove that the fixed point property is a topological property..

8. A space has "the fixed point property" if every continuous function f: X → Y leaves at least one fixed point, i.e. there exists such that x in X such that f(x)=x. Prove that the fixed point property is a topological property..

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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8. A space has "the fixed point property" if every continuous function f: X → Y leaves at
least one fixed point, i.e. there exists such that x in X such that f(x)=x. Prove that the
fixed point property is a topological property..

Expert Solution

Step 1: Given.

Given: 8. Suppose $X$ is any set and $f : X \to X$ is a continuous function. The point $x \in X$ is said to be fixed point of $f$ if $f (x) = x$ .

To show: Fixed point property is a topological property.

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