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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 is second-countable if there exists some countable collection of open subsets of such that any open subset of can be written as a union of elements of some subfamily of . 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 subsets that a space can have.

Properties
Second-countability is a stronger notion than 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 (has a countable dense subset) and 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 spaces, 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. In second-countable spaces—as in metric spaces—compactness, sequential compactness, and countable compactness are all equivalent properties. Urysohn's metrization theorem states that every second-countable, Hausdorff regular space is metrizable. It follows that every such space is 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 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 ==
Examples
• 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 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==
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