Talk:Invariant (mathematics)

I think that the definition of an invariant set is not correct. If I am not mistaken, a set $S$ is invariant under a map $T$ if $a\backslash in\; S\backslash Leftrightarrow\; T(a)\backslash in\; S$. In particular, if $T$ is a bijection (which is the typical case where this terminology is used), this means that $S=T(S)$, i.e. $S$ is a fixed element of the action of $T$ in the power set. I think the current definition corresponds to "closed": a set $S$ is closed under $T$ if $T(S)\backslash subseteq\; S$. Another common terminology is "fixed". A set $S$ is (pointwise) fixed by $T$ if $T(a)=a$ for all $a\backslash in\; S$, and setwise fixed if $T(S)=S$. So setwise fixed is a synonym for invariant (if $T$ is a bijection). However, these terms are sometimes confused: a good example is pointwise invariant and setwise invariant instead of pointwise fixed and setwise fixed. As mentioned above, another common term is "stable", which I believe is also a synonym for "invariant", and not for "closed". In any case, it might be good to find references to the different uses. arf