stable subspace

A subset $S$ of a larger set $T$ is said to a stable subset for a function $f:T\to T$ iff $f(S)\subset S$.
Alternative phrasings with the same meaning are:

• $f$ is an invariant subset for $f$

• $f$ stabilizes $S$

• $S$ is stable under (the action of) $f$

• $S$ is invariant under (the action of) $f$

• $S$ is left stable by/under $f$

• $S$ is left invariant by/under $f$