Normal covers of infinite products and normality of Σ-products

σ-Normality of Topological Spaces

Infinite Products of Infinite Measures

Let (Xi, Bi, mi) (i ∈ N) be a sequence of Borel measure spaces. There is a Borel measure μ on ∏ i∈N Xi such that if Ki ⊆ Xi is compact for all i ∈ N and ∏ i∈N mi(Ki) converges then μ( ∏ i∈N Ki) = ∏ i∈N mi(Ki)

On Cartesian Products of Orthogonal Double Covers

Let H be a graph on n vertices and G a collection of n subgraphs of H, one for each vertex, where G is an orthogonal double cover (ODC) of H if every edge of H occurs in exactly two members of G and any two members share an edge whenever the corresponding vertices are adjacent in H and share no edges whenever the corresponding vertices are nonadjacent in H. In this paper, we are concerned with ...

Infinite Sums, Infinite Products, and ζ(2k)

The most basic concept is that of an infinite sequence (of real or complex numbers in these notes). For p ∈ Z, let Np = {k ∈ Z : k ≥ p}. An infinite sequence of (complex) numbers is a function a : Np → C. Usually, for n ∈ Np we write a(n) = an, and denote the sequence by a = {an}n=p. The sequence {an}n=p is said to converge to the limit A ∈ C provided that for each > 0 there is an N ∈ Z such th...

Zeta regularization of infinite products

The problem of regularizing infinite products is a long standing one in mathematical and theoretical physics, as well as in mathematics, of course. Several either formal or rigorous techniques have been introduced. One that obtained an undiscussed favor is the zeta regularization. The idea is simple to write down in a formal way (so as it as been used for a long times by physicists), and has go...