6 Negative Dimension in General and Asymptotic Topology

We introduce the notion of negative topological dimension and the notion of weight for the asymptotic topological dimension. Quantizing of spaces of negative dimension is applied to linguistic statistics. Recently, Yu. I. Manin has presented his considerations for the density of spaces of negative dimension [1]. 1. Let us consider the simplest examples of (Haar) measures in the general case for the n-dimensional space. Let S n be an n-dimensional ball of radius r. In the spherical coordinates, the volume µ(S n) of the ball is equal to const 1 0 r n−1 dr = const r n. Here r n−1 stands for the density. In the sense of the Fourier transform, the multiplication by a coordinate is dual to the corresponding derivation. Therefore, we can speak of dual n times differentiable functions in the Sobolev space W n 2. Dirac distinguished between the left and right components " bra " and " ket " in the " bracket " inner product. The " dual " space of this space according to Dirac is the space W −n 2 of Sobolev distributions (generalized functions). In the same way we can define the functions in W s 2 by the " inner product, " where s is a positive noninteger number, and the space W −s 2 as the " inner product " conjugate to W s 2. One can similarly proceed with the density (or the weight) r s and r −s , by using, for instance, the Riesz kernel or the Bessel potential to represent functions in W s 2. Let us present an example of a space (of noninteger positive dimension) equipped with the Haar measure r σ , where 0 ≤ σ ≤ 1. On the closed interval 0 ≤ x ≤ 1 there is a scale 0 ≤ σ ≤ 1 of Cantor dust with the Haar measure equal to x σ for any interval (0, x) similar to the entire given set of the Cantor dust. The direct product of this scale by the Euclidean cube of dimension k − 1 gives the entire scale k + σ, where k ∈ Z and σ ∈ (0, 1). General definition of spaces of negative dimension. Let M t 0 be a compactum, of Hausdorff dimension t 0 , which is an element of a t-parameter scale of mutually embedded compacta, 0 < t < ∞. Two …