I have a doubt in writing a math proof:
Let $T: X \to X$ is continuous map and $A\subseteq X$. A set $A$ is $T$ invariant. Can we write it as $A$ is closed under the action of $T$? I am asking it because sometime we write in linear algebra that a subspace $A$ is closed under scalar multiplication so we can take the scalar multiplication as a map.
Yes, the phrase "$A$ is closed under the action of $T$" is fine and means that for all $a\in A$, $T(a)\in A$. Probably it would be more common to simply say "$A$ is closed under $T$" or "closed with respect to $T$", though.