product of a second-countable space is second-countable, although uncountable products need not be. The topology of a second-countable T1 space has cardinality less than or equal to c (the cardinality of the continuum). Any base for a second-countable space has a countable subfamily which is still a base. Every collection of disjoint open sets in a second-countable space is countable. Examples and counterexamples Consider the disjoint countable union . Define an equivalence relation and a quotient topology by identifying the left ends of the intervals - that is, identify 0 ~ 2 ~ 4 ~ … ~ 2k and so on. X is second-countable, as a countable union of second-countable spaces. However, X/~ is not first-countable at the coset of the identified points and hence also not second-countable. The above space is not homeomorphic to the same set of equivalence classes endowed with the obvious metric: i.e. regular Euclidean distance for two points in the same interval, and the sum of the distances to the left hand point for points not in the same interval -- yielding a strictly coarser topology than the above space. It is a separable metric space (consider the set of rational points), and hence is second-countable. The long line is not second-countable, but is first-countable. Notes References Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts. John G. Hocking and Gail S. Young (1961). Topology. Corrected reprint, Dover, 1988. General topology Properties of topological spaces