Given a property P \subseteq \Sigma ^\omega , the safety closure of P, written \overline {P}, is the smallest superset of P which is a safety property. It is the dual of the guarantee kernel, so \underline {P^\complement } = \overline {P}^\complement . This means that the safety closure of P contains all behaviours which cannot be shown to violate P by a finite bad prefix.
It follows that P is a safety property iff \overline {P} = P.
This is a closure operator, so it is extensive (\overline {P} \supseteq P), idempotent (\overline {\overline {P}} = \overline {P}), and monotonic (P \subseteq R implies \overline {P} \subseteq \overline {R}).
Topologically speaking, the safety closure is the (limit-)closure.
Intuitively, the safety closure of P is all behaviours for which any finite prefix extended in some way will get into P.