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| In [[topology]], a '''second-countable space''', also called a '''completely separable space''', is a [[topological space]] satisfying the '''second [[axiom of countability]]'''. A space is said to be second-countable if its topology has a [[countable]] [[base (topology)|base]]. More explicitly, this means that 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 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>. Like other countability axioms, the property of being second-countable restricts the number of [[open set]]s that a space can have.
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| Most "[[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 not countable, one can restrict to the set of all open balls with [[rational number|rational]] radii and whose centers have rational coordinates. This restricted set is countable and still forms a basis.
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| ==Properties==
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| 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.
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| 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. Therefore, the lower limit topology on the real line is not metrizable.
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| In second-countable spaces—as in metric spaces—[[compact space|compactness]], sequential compactness, and countable compactness are all equivalent properties.
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| [[Pavel Samuilovich Urysohn|Urysohn]]'s [[metrization theorem]] states that every second-countable, [[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.
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| ===Other properties===
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| *A continuous, [[open map|open]] [[image (mathematics)|image]] of a second-countable space is second-countable.
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| *Every [[subspace (topology)|subspace]] of a second-countable space is second-countable.
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| *[[Quotient space|Quotients]] of second-countable spaces need not be second-countable; however, ''open'' quotients always are.
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| *Any countable [[product space|product]] of a second-countable space is second-countable, although uncountable products need not be.
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| *The topology of a second-countable space has [[cardinality]] less than or equal to ''c'' (the [[cardinality of the continuum]]).
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| *Any base for a second-countable space has a countable subfamily which is still a base.
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| *Every collection of disjoint open sets in a second-countable space is countable.
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| == Examples ==
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| * Consider the disjoint countable union <math> X = [0,1] \cup [2,3] \cup [4,5] \cup \dotsb \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.
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| * Note that 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. It is a separable metric space (consider the set of rational points), and hence is second-countable.
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| * The [[long line (topology)|long line]] is '''not''' second countable.
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| ==References==
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| * Stephen Willard, ''General Topology'', (1970) Addison-Wesley Publishing Company, Reading Massachusetts.
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| * John G. Hocking and Gail S. Young (1961). ''Topology.'' Corrected reprint, Dover, 1988. ISBN 0-486-65676-4
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| [[Category:General topology]]
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| [[Category:Properties of topological spaces]]
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