Intopology, asecond-countable space, also called acompletely separable space, is atopological space whose topology has acountablebase. More explicitly, a topological space${\displaystyle T}$ is second-countable if there exists some countable collection${\displaystyle \mathcal{U}=\{U_i{\}}_{i=1}^{\mathrm{\infty }}}$ ofopen subsets of${\displaystyle T}$ such that any open subset of${\displaystyle T}$ can be written as a union of elements of some subfamily of${\displaystyle \mathcal{U}}$. A second-countable space is said to satisfy thesecond axiom of countability. Like othercountability axioms, the property of being second-countable restricts the number of open subsets that a space can have.

Many "well-behaved" spaces inmathematics are second-countable. For example,Euclidean space (Rⁿ) with its usual topology is second-countable. Although the usual base ofopen balls isuncountable, one can restrict this to the collection of all open balls withrational radii and whose centers have rational coordinates. This restricted collection is countable and still forms a basis.

Second-countability is a stronger notion thanfirst-countability. A space is first-countable if each point has a countablelocal base. Given a base for a topology and a pointx, the set of all basis sets containingx forms a local base atx. 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 uncountablediscrete space is first-countable but not second-countable.

Second-countability implies certain other topological properties. Specifically, every second-countable space isseparable (has a countabledense subset) andLindelöf (everyopen cover has a countable subcover). The reverse implications do not hold. For example, thelower limit topology on the real line is first-countable, separable, and Lindelöf, but not second-countable. Formetric 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,Hausdorffregular space ismetrizable. It follows that every such space iscompletely normal as well asparacompact. Second-countability is therefore a rather restrictive property on a topological space, requiring only a separation axiom to imply metrizability.

• A continuous,openimage of a second-countable space is second-countable.
• Everysubspace of a second-countable space is second-countable.
• Quotients of second-countable spaces need not be second-countable; however,open quotients always are.
• Any countableproduct of a second-countable space is second-countable, although uncountable products need not be.
• The topology of a second-countable T₁ space hascardinality less than or equal toc (thecardinality 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.

• Consider the disjoint countable union${\displaystyle X=[0,1]\cup [2,3]\cup [4,5]\cup \cdots \cup [2k,2k+1]\cup \cdots }$. Define an equivalence relation and aquotient 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.
• Thelong line is not second-countable, but is first-countable.

• 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

• General topology
• Properties of topological spaces

• Articles with short description
• Short description is different from Wikidata