Lebesgue's number lemma
Encyclopedia
In topology
, Lebesgue's number lemma, named after Henri Lebesgue
, is a useful tool in the study of compact
metric space
s. It states:
Such a number δ is called a Lebesgue number of this cover. The notion of a Lebesgue number itself is useful in other applications as well.
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...
, Lebesgue's number lemma, named after Henri Lebesgue
Henri Lebesgue
Henri Léon Lebesgue was a French mathematician most famous for his theory of integration, which was a generalization of the seventeenth century concept of integration—summing the area between an axis and the curve of a function defined for that axis...
, is a useful tool in the study of compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s. It states:
- If the metric space (X, d) is compact and an open cover of X is given, then there exists a number δ > 0 such that every subsetSubsetIn 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 X having diameterDiameterIn geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
less than δ is contained in some member of the cover.
Such a number δ is called a Lebesgue number of this cover. The notion of a Lebesgue number itself is useful in other applications as well.