  Flag varieties are well-known algebraic varieties with many important
geometric, combinatorial, and representation theoretic properties. A Hessenberg
variety is a subvariety of a flag variety identified by two parameters: an
element $X$ of the Lie algebra $\mathfrak{g}$ and a Hessenberg subspace
$H\subseteq \mathfrak{g}$. This paper considers when two Hessenberg spaces
define the same Hessenberg variety when paired with $X$. To answer this
question we present the containment poset $\mathcal{P}_X$ of type $A$
Hessenberg varieties with a fixed first parameter $X$ and prove directly that
if $X$ is not a multiple of the element $\bf 1$ then the Hessenberg spaces
containing the Borel subalgebra determine distinct Hessenberg varieties. Lastly
we give a natural involution on $\mathcal{P}_X$ that induces a homeomorphism of
varieties and prove additional properties of $\mathcal{P}_X$ when $X$ is a
regular nilpotent element.
