True /false : There exist $f : S^1 \rightarrow \mathbb{R}$ which is continious and onto

by jasmine   Last Updated August 14, 2019 07:20 AM

Is the following statement is true/false ?

There exist $$f : S^1 \rightarrow \mathbb{R}$$ which is continious and onto

my thinking : yes , because for every function $$f : S^1 \rightarrow \mathbb{R}$$ there exist uncountably many pairs of distinct points $$x$$ and $$y$$ $$\in S^1$$ such that $$f(x) = f(y)$$

Tags :

The statement is wrong. The image of a compact set under a continuous mapping is compact.

Martin R
August 14, 2019 07:16 AM

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