Equivalent Conditions for Element is Loop

Theorem
Let $M = \struct{S, \mathscr I}$ be a matroid.

Let $\mathscr B$ denote the set of all bases of $M$.

Let $x \in S$.


 * $(1)\quad x$ is a loop
 * $(2)\quad x \in \map \sigma \O$

where $\map \sigma \O$ denotes the closure of the empty set.
 * $(3)\quad \map \rho {\set x} = 0$

where $\rho$ denotes the rank function of $M$.
 * $(4)\quad \set x$ is a circuit
 * $(5)\quad \exists B \in \mathscr B : x \in B$ $x$ is not a loop