Lebesgue Covering Dimension

An important Dimension and one of the first dimensions investigated. It is defined in terms of covering sets, andis therefore also called the Covering Dimension. Another name for the Lebesgue covering dimension is theTopological Dimension.

A Space has Lebesgue covering dimension if for every open Cover of that space, there is an openCover that refines it such that the refinement has order at most . Consider how many elements of the covercontain a given point in a base space. If this has a maximum over all the points in the base space, then this maximum iscalled the order of the cover. If a Space does not have Lebesgue covering dimension for any , it is said tobe infinite dimensional.

Results of this definition are:

1. Two homeomorphic spaces have the same dimension,

2. has dimension ,

3. A Topological Space can be embedded as a closed subspace of a Euclidean Space Iffit is locally compact, Hausdorff, second countable, and is finite dimensional (in the sense of the Lebesgue Dimension), and

4. Every compact metrizable -dimensional Topological Space can be embedded in .

References

Dieudonne, J. A. A History of Algebraic and Differential Topology. Boston, MA: Birkhäuser, 1994.

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 414, 1980.

Munkres, J. R. Topology: A First Course. Englewood Cliffs, NJ: Prentice-Hall, 1975.

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