Yes. But here's a hint which gives the rough argument. Let f(x) = sin(1/x). Suppose (x(t),y(t)) is the path joining (X,f(X)) to (0,Y) so x(0) = X, y(0) = f(X), x(1) = 0 and y(1) = Y. As t goes to 1, x(t) goes to 0. But essentially, y(t) = f(x(t)) then has no limit so doesn't converge to Y. There are some technical details that need to be sorted out, but that's the basic idea.
Hello, When you prove that the path from x to z is transitive, can you really speak of a composition? I mean, as defined f:[0,1]->X will map either to x or y, what you would have to do is show that either [0,1] is homeomorphic to the set connecting f(0)=x to f(1)=y, or you could define a piecewise function g:[0,2]-->y such that g(x)={(f(x), if 0\leq x < 1) and (f'(x-1), if 1 \leq x < 2), where f:[0,1] --> X such taht f'(0)=x and f'(1)=y. Then since [0,2] is an interval of R, we can very easily prove it is connected. Am I mistaken? I apologise if this is completely wrong. Cheers!
"The proof that the Topologist's Sine Curve is not path connected is left as an exercise to the viewer."
Yes. But here's a hint which gives the rough argument. Let f(x) = sin(1/x). Suppose (x(t),y(t)) is the path joining (X,f(X)) to (0,Y) so x(0) = X, y(0) = f(X), x(1) = 0 and y(1) = Y. As t goes to 1, x(t) goes to 0. But essentially, y(t) = f(x(t)) then has no limit so doesn't converge to Y. There are some technical details that need to be sorted out, but that's the basic idea.
Waoo sir, sir m an Indian girl thank you
Hello,
When you prove that the path from x to z is transitive, can you really speak of a composition? I mean, as defined f:[0,1]->X will map either to x or y, what you would have to do is show that either [0,1] is homeomorphic to the set connecting f(0)=x to f(1)=y, or you could define a piecewise function g:[0,2]-->y such that g(x)={(f(x), if 0\leq x < 1) and (f'(x-1), if 1 \leq x < 2), where f:[0,1] --> X such taht f'(0)=x and f'(1)=y. Then since [0,2] is an interval of R, we can very easily prove it is connected. Am I mistaken?
I apologise if this is completely wrong.
Cheers!
I am from India. Your explanation is very easy to understand. Thanks for this video.
Great work
The wait is over
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