I am confused by an example of an embedding given in Lee's Introduction to Topological Manifolds.
If we let $a:[0,1)\rightarrow\mathbb{C}$ be given by $a(s)=e^{2\pi is}$, then $a$ is not homeomorphic onto its image for the following reason: $[0,1/2)$ is open in $X=[0,1)$, but its image is not open in the the image set $S^1$. This makes sense. However, the author states that if we restrict $a$ to any interval $[0,b)$ for $0<b<1$, then this is in fact an embedding. I am having trouble seeing this. If we let $b=1/2$, then $[0,1/4)$ is open in $[0,1/2)$, but its image is not open in $S^1$. Am I missing something?
If $b=\frac12$ then the image is a half circle, with open endpoint.
The image of $[0,\frac14)$ is open in that half-circle.