In Loring Tu's book introduction to manifolds, the definition of a local flow, page 223-224 we have:
For a smooth vector field $X$ on $M$ and $p$ in $M$, there exists an open set $U$ of $p$ and an $\epsilon>0$ and a smooth map $\varphi: (-\epsilon,\epsilon)\times U \rightarrow M$ such that
$\varphi_t(q)$ is an integral curve of $X$ at $q$, for all $q\in U$.
$\varphi_{s}\circ \varphi_{t}=\varphi_{s+t}$ whenever both sides are defined.
This definition implies then that
The map $\varphi_t:U\rightarrow \varphi_{t}(U)$ is a diffeomorphism onto its image.
My question is why is $\varphi_{t}(U)$ a manifold? I think it should be open, but I don't see how that follows from the definitions.
Best Answer
Let $\mathcal{D}_t$ denote those points for which the flow is defined for time at least $t$ ($0$ to $t$ or $t$ to $0$ depending on the sign, not necessarily $-t$ to $t$). One can show that $\mathcal{D}_t$ is open. Together with $\phi_{s+t} = \phi_s\phi_t$, the map $\phi_t\colon\mathcal{D}_t\to\mathcal{D}_{-t}$ is a diffeomorphism. The $U$ in question is an open subset of $\mathcal{D}_t$, hence $\phi_t(U)$ is open in $M$.
Here's a sketch of why $\mathcal{D}_t$ is open. Pick a $p\in\mathcal{D}_t$ and look at its flow on $[0, t]$. This is compact and one can construct a neighbourhood $W$ of this curve where the flow is defined for some tiny $\delta$. Within this $W$ there's a smaller neighbourhood $W_1$ of $p$ on which it makes sense to move by $\delta$, $n$ times i.e., composing $\phi_\delta$ $n$ times (to see this, observe that I can move $2\delta$ units in $W\cap \phi_\delta^{-1}W$) where $n$ is chosen so that $n\delta>t$. So, $W_1\subseteq\mathcal{D}_t$.
A reference for the above is Warner's Foundations of Differential Manifolds and Lie Groups. Alternatively, as mentioned in the comments, since $\phi_t$ is a homeomorphism, by invariance of domain, the image of the open set $U$ is open.