A representation of an inverse semigroup by means of partial open homeomorphisms of a topological \(T_0\)-space \(X\) is called topologically complete if the domains of these partial homeomophisms form a base of the topology of \(X\).
The author gives a method for constructing topologically complete representations by means of special ternary relations. This method makes it possible to obtain a pseudo-elementary axiomatization in \(T_1\), \(T_2\), and \(T_3\)-spaces. Also, the author proves that any antigroup has a natural topological structure \(\tau\) such that all of its faithful topologically complete representations are continuous, and \(\tau\) is the minimal topology with this property.