Lebesgue covering dimension explained
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in atopologically invariant way.[1] [2]
Informal discussion
For ordinary Euclidean spaces, the Lebesgue covering dimension is just the ordinary Euclidean dimension: zero for points, one for lines, two for planes, and so on. However, not all topological spaces have this kind of "obvious" dimension, and so a precise definition is needed in such cases. The definition proceeds by examining what happens when the space is covered by open sets.
In general, a topological space X can be covered by open sets, in that one can find a collection of open sets such that X lies inside of their union. The covering dimension is the smallest number n such that for every cover, there is a refinement in which every point in X lies in the intersection of no more than n + 1 covering sets. This is the gist of the formal definition below. The goal of the definition is to provide a number (an integer) that describes the space, and does not change as the space is continuously deformed; that is, a number that is invariant under homeomorphisms.
The general idea is illustrated in the diagrams below, which show a cover and refinements of a circle and a square.
| - | | The first image shows a refinement (on the bottom) of a colored cover (on the top) of a black circular line. Note how in the refinement, no point on the circle is contained in more than two sets, and also how the sets link to one another to form a "chain". |
| The top half of the second image shows a cover (colored) of a planar shape (dark), where all of the shape's points are contained in anywhere from one to all four of the cover's sets. The bottom illustrates that any attempt to refine said cover such that no point would be contained in more than two sets—ultimately fails at the intersection of set borders. Thus, a planar shape is not "webby": it cannot be covered with "chains", per se. Instead, it proves to be thicker in some sense. More rigorously put, its topological dimension must be greater than 1. | |
Formal definition
The first formal definition of covering dimension was given by Eduard Čech, based on an earlier result of Henri Lebesgue.[3]
A modern definition is as follows. An open cover of a topological space is a family of open sets such that their union is the whole space,
= . The
order or
ply of an open cover
= is the smallest number (if it exists) for which each point of the space belongs to at most open sets in the cover: in other words
1 ∩ ⋅⋅⋅ ∩
+1 =
for
1, ...,
+1 distinct. A refinement of an open cover
= is another open cover
=, such that each
is contained in some
. The
covering dimension of a topological space is defined to be the minimum value of such that every finite open cover
of
X has an open refinement
with order  + 1. The refinement
can always be chosen to be finite.
[4] Thus, if is finite,
1 ∩ ⋅⋅⋅ ∩
+2 =
for
1, ...,
+2 distinct. If no such minimal exists, the space is said to have infinite covering dimension.
As a special case, a non-empty topological space is zero-dimensional with respect to the covering dimension if every open cover of the space has a refinement consisting of disjoint open sets, meaning any point in the space is contained in exactly one open set of this refinement.
Examples
The empty set has covering dimension -1: for any open cover of the empty set, each point of the empty set is not contained in any element of the cover, so the order of any open cover is 0.
Any given open cover of the unit circle will have a refinement consisting of a collection of open arcs. The circle has dimension one, 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 two open 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, any open cover of the unit disk in the two-dimensional plane can be refined so that any point of the disk is contained in no more than three open sets, while two are in general not sufficient. The covering dimension of the disk is thus two.
has covering dimension
n.
Properties
if and only if for any
closed subset A of
X, if
is continuous, then there is an extension of
to
. Here,
is the
n-dimensional sphere.
- Ostrand's theorem on colored dimension. If is a normal topological space and
= is a locally finite cover of of order ≤ + 1, then, for each 1 ≤ ≤ + 1, there exists a family of pairwise disjoint open sets
= shrinking
, i.e.
, ⊆
, and together covering .
Relationships to other notions of dimension
- For a paracompact space, the covering dimension can be equivalently defined as the minimum value of, such that every open cover
of (of any size) has an open refinement
with order  + 1.
[5] In particular, this holds for all metric spaces.
is greater or equal to its
cohomological dimension (in the sense of
sheaves),
[6] that is, one has
for every sheaf
of abelian groups on
and every
larger than the covering dimension of
.
- In a metric space, one can strengthen the notion of the multiplicity of a cover: a cover has -multiplicity if every -ball intersects with at most sets in the cover. This idea leads to the definitions of the asymptotic dimension and Assouad–Nagata dimension of a space: a space with asymptotic dimension is -dimensional "at large scales", and a space with Assouad–Nagata dimension is -dimensional "at every scale".
See also
References
- Book: Edgar, Gerald A.. 2356043. Measure, topology, and fractal geometry. Second. Undergraduate Texts in Mathematics. Springer-Verlag. 2008. 978-0-387-74748-4. Topological Dimension. 85–114.
- Book: Engelking, Ryszard. Dimension theory. 0482697 . North-Holland Mathematical Library. 19. North-Holland. Amsterdam-Oxford-New York. 1978. 0-444-85176-3.
- Book: Godement . Roger . Roger Godement . Topologie algébrique et théorie des faisceaux . Hermann . Paris . 0102797 . 1958. fr . Publications de l'Institut de Mathématique de l'Université de Strasbourg. III.
- Book: Hurewicz, Witold. 0006493 . Wallman. Henry. Dimension Theory . Princeton Mathematical Series. 4 . Princeton University Press. 1941.
- Book: Munkres, James R. . James Munkres. Topology . 2000 . 2nd . Prentice-Hall . 0-13-181629-2. 3728284 .
- 0288741 . Ostrand. Phillip A.. Covering dimension in general spaces. General Topology and Appl.. 1 . 1971. 3. 209–221.
Further reading
Historical
- Karl Menger, General Spaces and Cartesian Spaces, (1926) Communications to the Amsterdam Academy of Sciences. English translation reprinted in Classics on Fractals, Gerald A.Edgar, editor, Addison-Wesley (1993)
- Karl Menger, Dimensionstheorie, (1928) B.G Teubner Publishers, Leipzig.
Modern
- Book: Pears, Alan R.. Dimension Theory of General Spaces. 1975. . 0-521-20515-8 . 0394604.
- V. V. Fedorchuk, The Fundamentals of Dimension Theory, appearing in Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I, (1993) A. V. Arkhangel'skii and L. S. Pontryagin (Eds.), Springer-Verlag, Berlin .
Notes and References
- Sur les correspondances entre les points de deux espaces. 2 . 1921. Henri. Lebesgue. Henri Lebesgue. Fundamenta Mathematicae. 256–285. 10.4064/fm-2-1-256-285. fr.
- The origins of the concept of dimension. 42. 1979. R.. Duda. Colloquium Mathematicum. 95–110. 10.4064/cm-42-1-95-110. 0567548. free.
- .
- Proposition 1.6.9 of Book: Engelking, Ryszard. Dimension theory. 0482697 . North-Holland Mathematical Library. 19. North-Holland. Amsterdam-Oxford-New York. 1978. 0-444-85176-3.
- Proposition 3.2.2 of Book: Engelking, Ryszard. Dimension theory. 0482697 . North-Holland Mathematical Library. 19. North-Holland. Amsterdam-Oxford-New York. 1978. 0-444-85176-3.
- Godement 1973, II.5.12, p. 236
