Is a set system an independence system, if and only if it is an accessible system, and has the Interval Property without Lower Bounds
Suppose that a set system $F$ is accessible and satisfies the interval property. We show that it is hereditary. Let $C = \{x_1,\ldots,x_n\}$ be any set. Without loss of generality, by accessibility we can assume that for all $t \in \{0,\ldots,n\}$, $\{x_1,\ldots,x_t\} \in F$. We show by induction that all subsets $B$ of $C$ are in $F$. If $|B| = 0$ then $B = \emptyset \in F$ (the case $t = 0$ above). Now suppose $B = \{x_{i_1}, \ldots, x_{i_k}\}$, where $i_1 &lt; i_2 &lt; \cdots &lt; i_k$. By assumption, $A = \{x_{i_1},\ldots,x_{i_{k-1}}\} \in F$. Now $A \subseteq \{x_1,\ldots,x_{i_k-1}\}$ and both $\{x_1,\ldots,x_{i_k-1}\},\{x_1,\ldots,x_{i_k}\} \in F$, and so by the interval property, $B = A \cup \{x_{i_k}\} \in F$.