# Does There exist a Continuous Surjection from $S^1$ to $[0, 1]$?

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?

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.

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 $$f(e^{ix})=\frac{1+\cos x}2$$