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Let $M$ be a smooth manifold. An open subset $D$ of $\mathbb{R}\times M$ is called a flow domain for $M$ if $\forall p\in M$, the set $$D^{(p)}=\{t\in\mathbb{R}:(t,p)\in D\}$$ is an open interval containing $0$. A smooth flow on $M$ is a smooth map $F$ from a flow domain $D$ to $M$ that satisfies: $\forall p\in M$, $$F(0,p)=p,$$ and for all $s\in D^{(p)}$ and $t\in D^{(F(s,p))}$ such that $s+t\in D^{(p)}$, $$F(t,F(s,p))=F(t+s,p).$$ Here comes the question. For any $p_0\in M$, why is $F(t,p)$ defined and smooth for all $(t,p)$ sufficiently close to $(0,p_0)$? This property is mentioned in Lee's ISM, and Lee said that it is because $D$ is open. But my knowledge of topology is unable to get me out of the question. Would you do me a favor? Thank you.

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    $\begingroup$ yes, this is a property of open sets in a metric space: each point of the set is the center of an $\epsilon$-ball which is contained in the open set $\endgroup$ Jul 29 '21 at 23:02
  • $\begingroup$ For $t$ near $0$, $(t,p_0)$ is, by definition, contained in $D$. How about $(0,p)$ or, more generally, $(t,p)$? $\endgroup$
    – Steve
    Jul 29 '21 at 23:18
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    $\begingroup$ Yes. Maybe it would clarify things if you stated, or tried to formalize what you mean by "near" in the case of $(t,p)$ being near $(0,p_0)$ $\endgroup$ Jul 29 '21 at 23:22
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    $\begingroup$ This is a basic fact about the product topology: every open neighborhood of $(x,y)\in X\times Y$ contains a subset of the form $U\times\{y\}$, where $U\subseteq X$ is an open neighborhood of $x\in X$. $\endgroup$
    – Kajelad
    Jul 29 '21 at 23:38
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I think I have found the answer. It is all about a basis for a topology. According to Munkres's book, the product topology on $\mathbb{R}\times M$ is the topology having as basis the collection of all sets of the form $U\times V$, where $U$ and $V$ are open sets in $\mathbb{R}$ and $M$, respectively. Now $(0,p_0)$ is in the open set $D$, so there exist open sets $U\subseteq\mathbb{R}$ and $V\subseteq M$ such that $(0,p_0)\in U\times V\subseteq D$. The rest is history.

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