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Obviously there is no homeomorphism between them.

As both of them compact and connected I am facing difficulty to show such continuous surjection not exist. Actually I have a feeling that there does not exist such continuous surjection but how?

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Hint: Consider the map $(\cos x, \sin x) \mapsto \cos x$, this is a continuous surjection onto $[-1,1]$. Since $[-1,1]$ is homeomorphic to $[0,1]$ via $x \mapsto \frac{1+x}{2}$, composition gives the desired map.

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If the interval had been $(0,1)$, there would be no surjection because any homeomorphism from a compact space is compact.

However, since the interval here is closed, we can simply project and obtain a surjection. If we write the points of $S^1$ as $e^{ix}$, then a surjection is given by $$f(e^{ix})=\frac{1+\cos x}2$$

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