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.[1] 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.