During my research, I came a cross on these notions :

Definition 1:A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary intersections, with $X \in \mathcal T$ and if $\mathcal U=\{ A \in \mathcal T\text{ | } A\neq X \text{ and } \forall C \in \mathcal T, A\subset C\text{ or } C \subset A \}$ then $O_{\mathcal T}= \bigcup \limits_{F \in \mathcal U} F$

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Definition 2:Let $U \subset X$ we denote $\langle U\rangle_S=\bigcap \limits_{F \in \mathcal T, U \subset F} F$

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Definition 3:We say that $S=(X,\mathcal T)$ a structure has a dimension if :$ \forall U \subset X,U\neq \emptyset, U \cap O_{\mathcal T}= \emptyset$ and $A=\langle U\rangle_S$,

with $\forall V \subset A, V \cap O_{\mathcal T}=\emptyset$ , $\text{card}(V)>\text{card}(U)$, $\exists v_o \in V $

such that $v_o \in \langle v \text{ | } v \in V \text{ and } v \neq v_o\rangle_S $

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Question:Is this generalization of the dimension already existing?

PS : in this case, a set $E$ has a dimension, with the structure $(E,\mathcal P(E) )$.