Given a ground set $E$, and a matroid closure operator $\tau$ on $\mathcal P(E)$, we can define a set system $(E,F)$ with $$ F := \{X \in \mathcal P(E): \forall x \in X, x \notin \tau(X-\{x\}) \}$$ which is a matroid.
A matroid closure operator is defined as a special closure operator in the sense for a Moore closure system, with the MacLane–Steinitz exchange property.
If $\tau$ is a closure operator rather than a matroid closure operator, what is name of the concept for the set system $(E,F)$ defined in the same way as above?
Thanks.