A set X is called totally disconnected if each of its members is its own connected component.
The most famous example of a totally disconnected set is the Cantor set. Note that it has the cardinality of the continuum ℵ=2ℵ0. Totally disconnected sets can be big, even in R! This set is homeomorphic to the topological space {0,1}N; in fact, every set FN with F finite is totally disconnected when equipped with the product topology.