What is the term for a transitive antisymmetric relation? I mean, a relation $\prec$ that satisfies
if $a\prec b$ and $b\prec c$, then $a\prec c$,
if $a\prec b$ and $b\prec a$, then $a = b$.
What is its place in Order Theory?
For example, the following relation $\prec$ on subsets of a topological space is transitive and antisymmetric, but generally is neither a partial order nor a strict partial order:
Let $X\prec Y$ mean that there is an open set $U$ such that $X\subset U\subset Y$.
(By $A\subset B$ I mean that $A$ is a subset of $B$. In particular, $A\subset A$ for all $A$.)
P.S. I thought of this relation on subsets of a topological space because if I define the analogues of such order-theoretic notions as supremum, directed set, and compact element, then compact subsets of a topological space will be exactly the compact elements of this relation.