Let $A$ be a set, and $T:A\rightarrow A$ a transformation of that set. We say that $x\in A$ is fixed by $T$ or that $T$ fixes $x$ whenever $$T(x)=x.$$ The subset of fixed elements is called the fixed set of $T$ , and is frequently denoted as $A^T$

We say that a subset $B\subset A$ is fixed by $T$ whenever all elements of $B$ are fixed by $T$ i.e. $$B\subset A^T.$$ If this is so, $T$ restricts to the identity transformation on $B$

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 $B\subset A$ is fixed, if it is fixed by all the elements of the family, i.e. whenever $$B\subset \bigcap_{i\in I} A^{T_i}.$$