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The form ω is called a local contact form. On the overlap of two such open sets, the local forms agree up to a non-vanishing scalar multiple. If the form can be globally defined, then M is a contact manifold. An example of a contact structure is given by the manifold of straight lines in the plane R2 . Local coordinates (x, y, z) are given by letting (x, y) be the point of contact and 0 < z < π the angle between the line and a horizontal line. ) The 1-form is given by ω(x, y, z) = sin(z)dx − cos(z)dy.

Assuming that we have defined all spaces of dimension ≤ n, we say that a space X has dimension ≤ n + 1 if, for any disjoint closed subsets C and D of X, there is a closed subset T of X with dimension T ≤ n such that X\T is the union of two disjoint open subsets, one containing C and the other containing D. We then say that a topological space has dimension n if it has dimension ≤ n but it does not have dimension ≤ n − 1. ) if [x, y] is an elementary interval in L, then d(y) = d(x)+ 1. dimension of a complex Let X be a CWcomplex and let E be the set of cells of X.

For example, if f = φ, the Euler phi function, and g = τ , the number of divisors function, disjoint sets Two sets X and Y which have no common elements. Symbolically, X and Y are disjoint if X∩Y =∅. (φ ∗ τ )(10) = φ(1)τ (10)+φ(2)τ (5)+φ(5)τ (2)+φ(10)τ (1) = 18 . In fact, φ ∗ τ = σ , the sum of divisors function. The Dirichlet convolution is also called the Dirichlet product. Dirichlet inverse The Dirichlet inverse of an arithmetic function f is a function f −1 such that the Dirichlet convolution f ∗f −1 = I , the identity function.