Second-countable space

In

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

Many "

radii and whose centers have rational coordinates. This restricted set is countable and still forms a basis.

Properties

Second-countability is a stronger notion than

is first-countable but not second-countable.

Second-countability implies certain other topological properties. Specifically, every second-countable space is

Therefore, the lower limit topology on the real line is not metrizable.

In second-countable spaces—as in metric spaces—compactness, sequential compactness, and countable compactness are all equivalent properties.

. Second-countability is therefore a rather restrictive property on a topological space, requiring only a separation axiom to imply metrizability.

Other properties

• A continuous,
  open image
  of a second-countable space is second-countable.
• Every
  subspace
  of a second-countable space is second-countable.
• Quotients of second-countable spaces need not be second-countable; however, open quotients always are.
• Any countable
  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

• Consider the disjoint countable union ${\displaystyle X=[0,1]\cup [2,3]\cup [4,5]\cup \dots \cup [2k,2k+1]\cup \dotsb }$. 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.
  ISBN 0-486-65676-4