Example of a continuous function from $[0,1]$ to a Hausdorff space $X$ that is not one-one but the image is homeomorphic to $[0,1]$.

Question

Suppose that $X$ is a Hausdorff space and $\alpha : [0, 1] \to X$ is a continuous function. If $\alpha$ is one-one, then prove that the image of $\alpha$ is homeomorphicto $[0, 1]$. Give an example where $\alpha$ is not one-one but the image of $\alpha$ is homeomorphic to $[0, 1]$.$ $

I am able to show the first part but not the second part. Any help would be highly appreciated!

Answer 1

Any non-injective surjective function between $[0,1]$ and $[0,1]$ should do the trick.

Answer 2

Consider for example $\alpha(x) = \sin(\pi x)$. Then $\alpha$ is not one-to-one, since $\alpha(0) = \alpha(1) = 0$, but $\alpha([0,1]) = [0,1]$, which is clearly homemorphic to $[0,1]$.

Answer 3

Try $f(x) = 0 \text{ for } 0 \le x \le \frac{1}{2}$ and $f(x) =2x -1 \text { for } \frac{1}{2} \le x \le 1$