Editing Second-countable space (section)

== Examples ==
* Consider the disjoint countable union <math> X = [0,1] \cup [2,3] \cup [4,5] \cup \dots \cup [2k, 2k+1] \cup \dotsb</math>. 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 (topology)|long line]] is not second-countable, but is first-countable.