In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. It is named for Ernst Leonard Lindelöf. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X... In mathematics the term countable set is used to describe the size of a set, e. ... Ernst Leonard Lindelöf, (7 March 1870–4 June 1946), Finnish topologist for whom Lindelöf space is named; son of Leonard Lorenz Lindelöf and brother of the philologist Uno Lorens Lindelöf. ...

A Lindelöf space is a generalization of the more commonly used notion of compactness, which requires that the subcover be finite. In fact, we can define A-compact, where A is a cardinal, if every open cover has a subcover of cardinality strictly less than A; then compact is -compact and Lindelöf is -compact. Several specialized usages of the terms compact and compactness exist. ... Alternative meaning: number of pitch classes in a set. ... In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X...

In general, no implications hold (in either direction) between the Lindelöf notion and other compactness notions, such as paracompact, (but regular Lindelöf implies paracompact, Morita Theorem) which are discussed in the compactness page. Any second-countable space is a Lindelöf space, but not conversely. Several specialized usages of the terms compact and compactness exist. ... In topology, a second-countable space is a topological space satisfying the second axiom of countability. Specifically, a space is said to be second-countable if its topology has a countable base. ...

However, the matter is simpler for metric spaces. A metric space is Lindelöf if and only if it is separable if and only if it is second-countable. In mathematics, a metric space is a set (or space) where a distance between points is defined. ... Separable can refer to: Separable space in topology Separable sigma algebra in measure theory Separable differential equations This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In topology, a second-countable space is a topological space satisfying the second axiom of countability. Specifically, a space is said to be second-countable if its topology has a countable base. ...

An open subspace of a Lindelöf space is not necessarily Lindelöf. However, a closed subspace must be Lindelöf.

Lindelöf is preserved by continuous maps. However, it is not necessarily preserved by products, not even by finite products.

Product of Lindelöf spaces

The product of Lindelöf spaces is not necessarily Lindelöf. The usual example of this is the Sorgenfrey plane S, which is the product of R under the half-open interval topology with itself. Open sets in the Sorgenfrey plane are unions of half-open rectangles that include the south and west edges and omit the north and east edges, including the northwest, northeast, and southeast corners. In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. ... In mathematics, the real line is simply the set of real numbers. ... In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

Consider the open covering of S which consists of: In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X...

1. The set of all points (x, y) with x < y
2. The set of all points (x, y) with x+1 > y
3. For each real x, the half-open rectangle [x, x+2) × [-x, -x+2)

The thing to notice here is that each rectangle [x, x+2) × [-x, -x+2) covers exactly one of the points on the line x = -y. None of the points on this line is included in any of the other sets in the cover, so there is no proper subcover of this cover, which therefore contains no countable subcover.

History

The Lindelöf space is named for Finnish mathematician Ernst Leonard Lindelöf. Ernst Leonard Lindelöf, (7 March 1870–4 June 1946), Finnish topologist for whom Lindelöf space is named; son of Leonard Lorenz Lindelöf and brother of the philologist Uno Lorens Lindelöf. ...

References

• Michael Gemignani, Elementary Topology (ISBN 0-486-66522-4) (see especially section 7.2)
• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology (ISBN 0-486-68735-X)