Second-countable space

In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space $T$ is second-countable if there exists some countable collection ${\mathcal {U}}=\{U_{i}\}_{i=1}^{\infty }$ of open subsets of $T$ such that any open subset of $T$ can be written as a union of elements of some subfamily of ${\mathcal {U}}$ . A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have.

Many "well-behaved" spaces in mathematics are second-countable. For example, Euclidean space (Rⁿ) with its usual topology is second-countable. Although the usual base of open balls is uncountable, one can restrict to the collection of all open balls with rational radii and whose centers have rational coordinates. This restricted set is countable and still forms a basis.

A continuous, image of a second-countable space is second-countable.

open

Every of a second-countable space is second-countable.

subspace

of second-countable spaces need not be second-countable; however, open quotients always are.

Quotients

Any countable of a second-countable space is second-countable, although uncountable products need not be.

product

The topology of a second-countable T₁ space has less than or equal to c (the cardinality of the continuum).

cardinality

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.

Consider the disjoint countable union $X=[0,1]\cup [2,3]\cup [4,5]\cup \dots \cup [2k,2k+1]\cup \dotsb$ . Define an equivalence relation and a 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.

quotient topology

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 is not second-countable, but is first-countable.

long line

Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.

John G. Hocking and Gail S. Young (1961). Topology. Corrected reprint, Dover, 1988. 0-486-65676-4