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 sets that a space can have.

Many "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 balls is uncountable, one can restrict to the collection of all open balls with rational radii and whose centers have rational coordinates. This restricted set is countable and still forms a basis.

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.

- 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 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.

- Consider the disjoint countable union . 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 weaker 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 it is first-countable.

- ↑ Willard, theorem 16.11, p. 112

In topology and related areas of mathematics, a **metrizable space** is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metric

In mathematics, a topological space is called **separable** if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In mathematics, a **paracompact space** is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by Dieudonné (1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.

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In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

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In mathematics, a **Lindelöf space** is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of *compactness*, which requires the existence of a *finite* subcover.

In mathematics, the **lower limit topology** or **right half-open interval topology** is a topology defined on the set of real numbers; it is different from the standard topology on and has a number of interesting properties. It is the topology generated by the basis of all half-open intervals [*a*,*b*), where *a* and *b* are real numbers.

In topology, the **long line** is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore, it serves as one of the basic counterexamples of topology. Intuitively, the usual real-number line consists of a countable number of line segments [0, 1) laid end-to-end, whereas the long line is constructed from an uncountable number of such segments.

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In the mathematical discipline of general topology, a **Polish space** is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.

In topology, a branch of mathematics, a **first-countable space** is a topological space satisfying the "first axiom of countability". Specifically, a space *X* is said to be first-countable if each point has a countable neighbourhood basis. That is, for each point *x* in *X* there exists a sequence *N*_{1}, *N*_{2}, … of neighbourhoods of *x* such that for any neighbourhood *N* of *x* there exists an integer *i* with *N*_{i} contained in *N*. Since every neighborhood of any point contains an open neighborhood of that point the neighbourhood basis can be chosen without loss of generality to consist of open neighborhoods.

In topology and related areas of mathematics, a **subspace** of a topological space *X* is a subset *S* of *X* which is equipped with a topology induced from that of *X* called the **subspace topology**.

* Counterexamples in Topology* is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr.

In topology and related areas of mathematics, a **topological property** or **topological invariant** is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space *X* possesses that property every space homeomorphic to *X* possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In topology, a branch of mathematics, a **topological manifold** is a topological space which locally resembles real *n*-dimensional space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. All manifolds are topological manifolds by definition, but many manifolds may be equipped with additional structure. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" any additional structure the manifold has.

In topology and related fields of mathematics, a **sequential space** is a topological space that satisfies a very weak axiom of countability.

In mathematics, the **particular point topology** is a topology where a set is open if it contains a particular point of the topological space. Formally, let *X* be any set and *p* ∈ *X*. The collection

- 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

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