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Let ƒ : R¹ → Rk be a continuous function. Which of the following claims are true? Justify your answers give a counterexample (i) For every closed set X CR, the set f-¹(X) CR¹ is closed. (ii) For every closed set X CR", the set f(X) C Rk is closed. (iii) For every sequentially compact set KC Rk, the set f−¹(K) C R” is se- quentially compact. (iv) For every sequentially compact set K CR¹, the set f(K) C Rk is sequen- tially compact.

Let ƒ : R¹ → Rk be a continuous function. Which of the following claims are true? Justify your answers give a counterexample (i) For every closed set X CR, the set f-¹(X) CR¹ is closed. (ii) For every closed set X CR", the set f(X) C Rk is closed. (iii) For every sequentially compact set KC Rk, the set f−¹(K) C R” is se- quentially compact. (iv) For every sequentially compact set K CR¹, the set f(K) C Rk is sequen- tially compact.

Advanced Engineering Mathematics
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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Let f: R → Rk be a continuous function. Which of the following claims are true? Justify your answers give a counterexample (i) For every closed set X CR, the set f−¹(X) C Rn is closed. (ii) For every closed set X CR, the set f(X) C Rk is closed. (iii) For every sequentially compact set KC Rk, the set f-¹(K) C Rª is se- quentially compact. (iv) For every sequentially compact set KCR, the set f(K) C Rk is sequen- tially compact.
Transcribed Image Text:Let f: R → Rk be a continuous function. Which of the following claims are true? Justify your answers give a counterexample (i) For every closed set X CR, the set f−¹(X) C Rn is closed. (ii) For every closed set X CR, the set f(X) C Rk is closed. (iii) For every sequentially compact set KC Rk, the set f-¹(K) C Rª is se- quentially compact. (iv) For every sequentially compact set KCR, the set f(K) C Rk is sequen- tially compact.
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