Ε-net (metric spaces)
Encyclopedia
In the mathematics
of metric spaces, an
-net, or epsilon net relates to having a set which approximates another in the sense that any point in the second set is within a small distance of a point in the first (approximating) set.
such that each point of the space is within distance of some point in the set. For example, if , then the integers are an -net of the real numbers.
This should not be confused with the concept of a net
, which is equivalent to a sequence
in a metric space.
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...
of metric spaces, an
-net, or epsilon net relates to having a set which approximates another in the sense that any point in the second set is within a small distance of a point in the first (approximating) set.Definition
An -net is a set of points in a metric spaceMetric 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...
such that each point of the space is within distance of some point in the set. For example, if , then the integers are an -net of the real numbers.
This should not be confused with the concept of a net
Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the range of this function is...
, which is equivalent to a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
in a metric space.