  We develop notions of a representation of a topological graph E and of a
covariant representation of a topological graph E which do not require the
machinery of C*-correspondences and Cuntz-Pimsner algebras. We show that the
C*-algebra generated by a universal representation of E coincides with the
Toeplitz algebra of Katsura's topological-graph bimodule, and that the
C*-algebra generated by a universal covariant representation of E coincides
with Katsura's topological graph C*-algebra. We exhibit our results by
constructing the isomorphism between the C*-algebra of a row-finite directed
graph E with no sources and the C*-algebra of the topological graph arising
from the shift map acting on infinite path space E^\infty.
