The article examines the topological properties of the Zorgefrey line - a set of real numbers  assigned to the topology of the lower half-intervals of the form , and also considers its modifications. The conditions for the homeomorphism of the Zorgephreesian line with the Euclidean line and other topological spaces are analyzed. It is shown that, despite the coincidence of the carrier, the Zorgefrye line is not a homeomorphic standard real line due to the differences in the bases of topologies and their invariant properties.