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(iii) Let X be a set with at least two points. Prove that the discrete space (X, T) and the indiscrete space (X, T') do not have the fixed-point property.

(iii) Let X be a set with at least two points. Prove that the discrete space (X, T) and the indiscrete space (X, T') do not have the fixed-point property.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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part iii
(ii) A topological space (X, T) is said to have the fixed point property if every continuous mapping of (X, T) into itself has a fixed point. Show that the only intervals in R having the fixed point property are the closed intervals. (iii) Let X be a set with at least two points. Prove that the discrete space (X, T) and the indiscrete space (X, T') do not have the fixed-point property. (iv) Does a space which has the finite-closed topology have the fixed-point property? (v) Prove that if the space (X, T) has the fixed-point property and (Y, T1) is a space homeomorphic to (X, T), then (Y.T1) has the fixed-point property.
Transcribed Image Text:(ii) A topological space (X, T) is said to have the fixed point property if every continuous mapping of (X, T) into itself has a fixed point. Show that the only intervals in R having the fixed point property are the closed intervals. (iii) Let X be a set with at least two points. Prove that the discrete space (X, T) and the indiscrete space (X, T') do not have the fixed-point property. (iv) Does a space which has the finite-closed topology have the fixed-point property? (v) Prove that if the space (X, T) has the fixed-point property and (Y, T1) is a space homeomorphic to (X, T), then (Y.T1) has the fixed-point property.
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