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A topological space X is said to be maximal if its topology is maximal among all _T1 topologies over X without isolated points. It is known that a space is maximal if, and only if, it is extremely disconnected, nodec and every open set is irresolvable. We present some results about the complexity of those properties on countable spaces. A countable topological space X is analytic if its topology is an analytic subset of P(X) identified with the Cantor cube ^{0,1}X. No extremely disconnected space can be analytic and every analytic space is hereditarily resolvable. However, we construct an example of a nodec regular analytic space. © 2014 Elsevier B.V.