I'm working out of Robinson's Dynamical Systems textbook. In chapter 10 (p404) he defines a trapping region (or isolating neighborhood by Conley) for a flow $\varphi ^t$ on a metric space $M$ as a set $U$ such that $U$ is positively invariant and there is a $T>0$ such that $\varphi ^T(cl(U))\subset int(U)$
He then defines $A$ as an attracting set for a flow providing that there exists a trapping region $U$ such that $A=\cap _{t\ge 0}\varphi ^t(U)$
Then in proposition 1.2(a) He says that these are closed. The proof provided is as follows: Because of the property that $\varphi ^T(cl(U))\subset int(U)$, it follows that $A$ is also the intersection of closed sets and so is closed. $A=\cap _{t\ge 0}\varphi ^t(cl(U))$
I don't follow this argument at all. Why did this property allow him to replace in the definition of the attracting set the $\varphi ^t(U)$'s with $\varphi^t(cl(U))$'s? From there, he gets those are closed because a flow is a homeomorphism for each $t$ so thus maps closed sets to closed sets, but I'm trying to figure out how he made that first jump
I am not familiar with the flows, but I guess that the idea is similar to the following simple fact. Let $M$ be set $M$, $\varphi:M\to M$ be a map, and $U\subset V\subset M$ be sets such that $\varphi^T(V)\subset U$ for some natural $T$. Then $$\bigcap_{t\in\Bbb N }\varphi^t(U)= \bigcap_{t\in\Bbb N }\varphi^t(V).$$ Indeed, the inclusion $\subset $ is obvious. To prove the converse inclusion, let $x$ be any element of
$\bigcap_{t\in\Bbb N }\varphi^t(V)$ and $t\in\Bbb N$ be any number. Then $x\in \varphi^{t+T}(V)\subset \varphi^{t}(U)$. Thus $x\in \bigcap_{t\in\Bbb N }\varphi^t(U)$.