Let $\alpha : R \to R^2$ be the map$ \alpha(x)=(x,x^2)$ ;let M be the image set of $\alpha$. Show that M is a 1-manifold in $R^2$ covered by the single coordinate patch $\alpha$.
I know the definition of manifold, but I am not clear how to show that $\alpha^{-1}$ is continuous.
Besides, if I should prove “for each p \in M, there is an open set V of M containing p”? And how can I show this?
I am so sorry as I am a beginning learner.
The inverse of $\alpha$ is simply the projection to the first coordinate $(x, x^2) \mapsto x$. Specifically, it is the composition of continuous functions $M\hookrightarrow \Bbb R^2 \to \Bbb R$.
For every point $p\in M$, we need to find a chart whose domain contains $p$. (Note that by definition, the domain of a chart will be an open set in the manifold.)
In this example, it's trivial, as we have only one chart here, with the full domain $M$.
