I worked this theory : https://mathoverflow.net/questions/317668/a-new-generalization-of-the-dimension

For have a theorem of the dimension, and be more generaler and simpler  than the Matroïds.

> **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 $X \in  \mathcal T$.

>

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

>

>**Definition 3:** $S=(X,\mathcal T)$ a structure, we say the set $U\neq \emptyset$ is free if :

> $\forall u \in U, u \notin  \langle v \mid v \in U,v\neq u \rangle_S$  

>
> **Definition 4:** $S=(X,\mathcal T)$ a structure, we say this structur have a dimension if:
  
>$\forall U \subset X$ free, with $v \notin \langle U\rangle$, $U \cup \{v\}$ is free.

>

> **Definition 5:** $S=(X,\mathcal T)$ a structure, $F \in \mathcal T$, we note $\dim(F)=n$ if:

>the largest free set of $F$ have a cardinal of $n$.

>

> **Theorem:** $S=(X,\mathcal T)$ a structure with a dimension, $E,F \in \mathcal T$, 

> if $\dim(E)=\dim(F)<\infty$ and $E \subset F$ then $E=F$

>

*Question: Is this generalization of the dimension already existing?*