Editing Second-countable space

{{short description|Topological space whose topology has a countable base}}

In [[topology]], a '''second-countable space''', also called a '''completely separable space''', is a [[topological space]] whose topology has a [[countable]] [[base (topology)|base]]. More explicitly, a topological space <math>T</math> is second-countable if there exists some countable collection <math>\mathcal{U} = \{U_i\}_{i=1}^{\infty}</math> of [[open set|open]] subsets of <math>T</math> such that any open subset of <math>T</math> can be written as a union of elements of some subfamily of <math>\mathcal{U}</math>. A second-countable space is said to satisfy the '''second axiom of countability'''. Like other [[countability axiom]]s, the property of being second-countable restricts the number of open subsets that a space can have. 

Many "[[well-behaved]]" spaces in [[mathematics]] are second-countable. For example, [[Euclidean space]] ('''R'''<sup>''n''</sup>) with its usual topology is second-countable. Although the usual base of [[open ball]]s is [[uncountable]], one can restrict this to the collection of all open balls with [[rational number|rational]] radii and whose centers have rational coordinates. This restricted collection is countable and still forms a basis.

==Properties==

Second-countability is a stronger notion than [[first-countable space|first-countability]]. A space is first-countable if each point has a countable [[local base]]. Given a base for a topology and a point ''x'', the set of all basis sets containing ''x'' forms a local base at ''x''. Thus, if one has a countable base for a topology then one has a countable local base at every point, and hence every second-countable space is also a first-countable space. However any uncountable [[discrete space]] is first-countable but not second-countable.

Second-countability implies certain other topological properties. Specifically, every second-countable space is [[separable space|separable]] (has a countable [[dense (topology)|dense]] subset) and [[Lindelöf space|Lindelöf]] (every [[open cover]] has a countable subcover). The reverse implications do not hold. For example, the [[lower limit topology]] on the real line is first-countable, separable, and Lindelöf, but not second-countable. For [[metric space]]s, however, the properties of being second-countable, separable, and Lindelöf are all equivalent.<ref>Willard, theorem 16.11, p. 112</ref>  Therefore, the lower limit topology on the real line is not metrizable.

In second-countable spaces&mdash;as in metric spaces&mdash;[[compact space|compactness]], sequential compactness, and countable compactness are all equivalent properties.

[[Urysohn's metrization theorem]] states that every second-countable, [[Hausdorff space|Hausdorff]] [[regular space]] is [[metrizable]]. It follows that every such space is [[completely normal space|completely normal]] as well as [[paracompact]]. 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 map|open]] [[image (mathematics)|image]] of a second-countable space is second-countable.
*Every [[subspace (topology)|subspace]] of a second-countable space is second-countable.
*[[Quotient space (topology)|Quotients]] of second-countable spaces need not be second-countable; however, ''open'' quotients always are.
*Any countable [[product space|product]] of a second-countable space is second-countable, although uncountable products need not be.
*The topology of a second-countable T<sub>1</sub> 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 <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.

==Notes==
{{reflist}}

==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}}

{{Topology}}

[[Category:General topology]]
[[Category:Properties of topological spaces]]