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Problem 5. Give an example where X is not path connected, f: X→ Y is continuous, and im(f) is path connected. [Don't just draw a picture, specify the function explicitly.] lo exina getrollo et coo

Problem 5. Give an example where X is not path connected, f: X→ Y is continuous, and im(f) is path connected. [Don't just draw a picture, specify the function explicitly.] lo exina getrollo et coo

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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Problem 5. Give an example where X is not path connected, f: X→ Y is continuous, and im(f) is path connected. [Don't just draw a picture, specify the function explicitly.] e lo ring gewollt eit neovisd portatil Jibosda .01
Transcribed Image Text:Problem 5. Give an example where X is not path connected, f: X→ Y is continuous, and im(f) is path connected. [Don't just draw a picture, specify the function explicitly.] e lo ring gewollt eit neovisd portatil Jibosda .01
Expert Solution
Step 1

Actually, the the difficult part is not to find such function f but to find such space X. Here I have given one example. 

Take  X  be the topologist's sign curve (consider this as a subspace of R^2) 

i.e.  X = {(0,y) : -1<=y<=1}  U  {(x, sin(1/x)) : 0<x<=1} 

This X is a famous example of a space which is connected but not path connected. 

Now,  Take  Y = R

 

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