Everything about Lebesgue Covering Dimension totally explained
In
mathematics, the
Lebesgue covering dimension or
topological dimension of a
topological space is defined to be the minimum value of
n, such that every
open cover has an open
refinement in which no point is included in more than
n+1 elements. If no such minimal
n exists, the space is said to be infinite dimensional. This notion of dimension is named after
Henri Lebesgue, although it was independently arrived at by a number of contemporaneous mathematicians.
For example, consider some arbitrary open cover of the
unit circle. This open cover will have a refinement consisting of a collection of open arcs. The circle has dimension 1, by this definition, because any such cover can be further refined to the stage where a given point
x of the circle is contained in
at most 2 arcs. That is, whatever collection of arcs we begin with, some can be discarded or shrunk, such that the remainder still covers the circle, but with simple overlaps.
Similarly, consider the
unit disk in the two-dimensional
plane. It isn't hard to visualize that any open cover can be refined so that any point of the disk is contained in no more than three sets.
The Lebesgue covering dimension coincides with the
affine dimension of a finite
simplicial complex; this is the
Lebesgue covering theorem.
Some Properties
The covering dimension of a
normal space is less than or equal to the large
inductive dimension.
Suppose that the covering dimension of a space
X is less than or equal to
n and
A is a closed subset of
X.
If
is continuous, then there's an extension of
to
.
Here,
is the
n dimensional sphere.
History
The idea of topological dimension first became a topic of considerable interest in the early 20th century. The core ideas were independently arrived at and published by
Karl Menger,
L. E. J. Brouwer,
Pavel Urysohn and
Henri Lebesgue.
Further Information
Get more info on 'Lebesgue Covering Dimension'.
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