Let $A$ be a set, and $T:A\rightarrow A$ a transformation of that set. We say that $x\in A$ is an invariant of $T$ whenever $x$ is fixed by $T$ $$T(x)=x.$$ We say that a subset $B\subset A$ is invariant with respect to $T$ whenever $$T(B)\subset B.$$ If this is so, the restriction of $T$ is a well-defined transformation of the invariant subset: $$T\Big|_B : B\rightarrow B.$$ The definition generalizes readily to a family of transformations with common domain $$T_i : A\rightarrow A,\quad i\in I$$ In this case we say that a subset is invariant, if it is invariant with respect to all elements of the family.