Properties
- Homeomorphic spaces have the same covering dimension.
- The Lebesgue covering dimension coincides with the affine dimension of a finite simplicial complex; this is the Lebesgue covering theorem.
- The covering dimension of a normal space is less than or equal to the large inductive dimension.
- Covering dimension of a normal space X is 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.) A normal space satisfies the inequality if and only if for every locally finite open cover of the space there exists an open cover of the space which can be represented as the union of families, where, such that each contains disjoint sets and for each and .
Read more about this topic: Lebesgue Covering Dimension
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)