In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Lindelöf's lemma is a simple but useful lemma

Lemma (mathematics)

In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself...

 in topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 on the real line

Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

, named for the Finnish

Finland

Finland , officially the Republic of Finland, is a Nordic country situated in the Fennoscandian region of Northern Europe. It is bordered by Sweden in the west, Norway in the north and Russia in the east, while Estonia lies to its south across the Gulf of Finland.Around 5.4 million people reside...

 mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 Ernst Leonard Lindelöf

Ernst Leonard Lindelöf

Ernst Leonard Lindelöf, was a Finnish topologist after whom Lindelöf spaces are named; he was the son of Leonard Lorenz Lindelöf and brother of the philologist Uno Lorenz Lindelöf....

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Statement of the lemma

Let the real line have its standard topology. Then every open

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

 subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

 of the real line is a countable

Countable set

In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...

 union

Union (set theory)

In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...

 of open interval

Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

s.

Generalization

Lindelöf's lemma is also known as the statement that every open cover in a second-countable space

Second-countable space

In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...

 has a countable subcover

Cover (topology)

In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset. Formally, ifC = \lbrace U_\alpha: \alpha \in A\rbrace...

 (Kelley 1955:49) This means that every second-countable space

Second-countable space

In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...

 is also a Lindelöf space

Lindelöf space

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....

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