Formal Definition
We want the dimension of a point to be 0, and a point has empty boundary, so we start with
Then inductively, ind(X) is the smallest n such that, for every and every open set U containing x, there is an open V containing x, where the closure of V is a subset of U, such that the boundary of V has small inductive dimension less than or equal to n − 1. (In the case above, where X is Euclidean n-dimensional space, V will be chosen to be an n-dimensional ball centered at x.)
For the large inductive dimension, we restrict the choice of V still further; Ind(X) is the smallest n such that, for every closed subset F of every open subset U of X, there is an open V in between (that is, F is a subset of V and the closure of V is a subset of U), such that the boundary of V has large inductive dimension less than or equal to n − 1.
Read more about this topic: Inductive Dimension
Famous quotes containing the words formal and/or definition:
“The spiritual kinship between Lincoln and Whitman was founded upon their Americanism, their essential Westernism. Whitman had grown up without much formal education; Lincoln had scarcely any education. One had become the notable poet of the day; one the orator of the Gettsyburg Address. It was inevitable that Whitman as a poet should turn with a feeling of kinship to Lincoln, and even without any association or contact feel that Lincoln was his.”
—Edgar Lee Masters (1869–1950)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)