Talk:Invariant (mathematics)

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Invariant or closed

I think that the definition of an invariant set is not correct. If I am not mistaken, a set ${\displaystyle S}$  is invariant under a map ${\displaystyle T}$  if ${\displaystyle a\in S\Leftrightarrow T(a)\in S}$ . In particular, if ${\displaystyle T}$  is a bijection (which is the typical case where this terminology is used), this means that ${\displaystyle S=T(S)}$ , i.e. ${\displaystyle S}$  is a fixed element of the action of ${\displaystyle T}$  in the power set. I think the current definition corresponds to "closed": a set ${\displaystyle S}$  is closed under ${\displaystyle T}$  if ${\displaystyle T(S)\subseteq S}$ . Another common terminology is "fixed". A set ${\displaystyle S}$  is (pointwise) fixed by ${\displaystyle T}$  if ${\displaystyle T(a)=a}$  for all ${\displaystyle a\in S}$ , and setwise fixed if ${\displaystyle T(S)=S}$ . So setwise fixed is a synonym for invariant (if ${\displaystyle T}$  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