| Second-countable Space |
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Information AboutSecond-countable Space |
| CATEGORIES ABOUT SECOND-COUNTABLE SPACE | |
| general topology | |
| properties of topological spaces | |
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Most " Well-behaved " spaces in Mathematics are second-countable. For example, Euclidean Space (R''n'') 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 radii and whose centers have rational coordinates. This restricted set is countable and still forms a base. PROPERTIES Second-countability is a stronger notion than First-countability . Recall that 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 clearly has a countable local base at every point. 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 Space s, however, the properties of being second-countable, separable, and Lindelöf are all equivalent. 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, Regular Hausdorff 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'':
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