invariant

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.