In the theory of Borel equivalence relations under Borel reducibility (see [this MO answer](https://mathoverflow.net/questions/10481/when-is-a-classification-problem-wild/10494#10494), also [Greg Hjorth's excellent survey article](http://www.math.ucla.edu/~greg/223b.1.06s/hjorth.dvi)), which is concerned with general questions surrounding the relative difficulty of isomorphism relations and other relations arising in mathematics, particularly of the difficulty of their classification problems---the theory ultimately arranges these relations into a complex hierarchy under Borel reducibility---the class of countable graphs $\Gamma=\langle\mathbb{N},E\rangle$ is considered by identifying the graph $\Gamma$ with its edge relation $E\in 2^{\mathbb{N}^2}$, and using the ordinary product topology, and is thereby realized as a standard Borel space, which enjoys various uniqueness properties. See for example page 2 of [these notes by Simon Thomas and Scott Schneider](http://www.math.cmu.edu/~eschimme/Appalachian/ThomasNotes.pdf). The basic open sets of this topology on the space of countable graphs are therefore determined by specifying finitely many edges and finitely many non-edges. 

(Note, however, that for the purpose of the Borel reducibility theory, the principal focus is on the resulting $\sigma$-algebra of Borel sets, rather than on the topology of open sets.)

The isomorphism relation on all countable graphs is a complete analytic relation, rather than Borel, but when you restrict to the case of locally finite graphs, then it becomes a Borel relation, and these exhibit various universal and completeness properties in the hierarchy.