Metric spaces

If we give E the discrete topology and T the product topology, then T becomes a compact metric space homeomorphic to the Cantor set under the metric... 

A partial analogy between the dynamics of CA and the behaviors of continuous dynamical systems may be obtained by exploiting a fundamental property of CA systems namely, continuity in the Cantor-set Topology. We recall from section 2.2.1 that the collection of all one-dimensional configurations, or the CA phase space, r = where E = 0,1,..., fc 9 cr and Z is the set of integers by which each site of the lattice is indexed, is a compact metric space homeomorphic to the Cantor set under the metric... 

These three properties, (i)-(iii), are among the properties of open sets in a metric space. 

Figure 4 The path traversed in structural order-metric space as liquid water (SPC/E) is compressed isothermally at two different temperatures. Filled diamonds represent T = 260 K, and open triangles represent T = 400 K. The arrows indicate the direction of increasing density. A and C are states of maximum tetrahedral order at the respective temperatures, whereas B is a state of minimum translational order. Reprinted with permission from Ref. 29. 

Tashmetov, U. (1974) Connectivity of hyperspaces. Doklady Akademii Nauk SSSR. 215(2) 286-88. (Results regarding connected and locally connected compacts in a hyperspace are extended to the case of arbitrary, full metric spaces.)... 

The criteria (9.23), (9.24), and (9.26) are all rather obvious properties of Euclidean geometry. All of these properties can be traced back to mathematical properties of the scalar product (R Ry), the key structure-maker of a metric space. We therefore wish to determine whether a proposed definition of scalar product satisfies these criteria, and thus guarantees that M is a Euclidean space. 

Having made this long detour into vector geometry and metric spaces, the student of thermodynamics will naturally be impatient to learn the missing link that connects these disparate domains, i.e., that associates the scalar products of the geometry domain... 

The Hessian of the internal energy is the Gramian of a metric space. 

While such results can also be inferred from the classical formalism [F. H. Crawford. Phys. Rev. 72, 521A (1947)], they have a particularly transparent basis in the metric space Ms-... 

GENERAL TRANSFORMATION THEORY IN THERMODYNAMIC METRIC SPACE 357... 

It is a remarkable fact that properties (13.4a-c) are necessary and sufficient to give Euclidean geometry. In other words, if any rule can be found that associates a number (say, (X Y)) with each pair of abstract objects ( vectors X), Y)) of the manifold in a way that satisfies (13.4a-c), then the manifold is isomorphic to a corresponding Euclidean vector space. We introduced a rather unconventional rule for the scalar products (X Y) [recognizing that (13.4a-c) are guaranteed by the laws of thermodynamics] to construct the abstract Euclidean metric space Ms for an equilibrium state of a system S, characterized by a metric matrix M. This geometry allows the thermodynamic state description to be considerably simplified, as demonstrated in Chapters 9-12. 

This can be briefly explained as follows. Ordinary physical space is a metric 3-space, which means that it is a three-dimensional space within which we can perform measurements of distance and displacement. Very little thought convinces us that our concepts of a physical metric space are irreducibly connected to its matter content—that is, all our notions of distance and displacement are meaningless except insofar as they are defined as relations between objects. Similarly, all our concepts of physical time are irreducibly connected to the notion of material process. Consequently, it is impossible for us to conceive of physical models of metric spacetime without simultaneously imagining a universe of material and material process. From this, it seems clear to me that any theory that allows an internally self-consistent discussion of an empty metric spacetime is a deeply nonphysical theory. Since general relativity is exactly such a theory, it is fundamentally flawed, according to this view. 

Frechet [63] made an abstract formulation of the notion of distance in 1906. Hausdorff [64] proposed the term metric space, where he introduced the function d that assigns a nonnegative real number d p, q) (the distance between p and q) to every pair ip. q) of elements (points) of a nonempty set S. A metric space is a pair (S,d) if the function d satisfies several conditions, such as triangle inequality. In 1942, Menger [65] proposed that if we replace d(p, q) by a real function Fpq whose value is Fpq(x) for any real number x, this can be interpreted as the probability that the distance between p and q is less than x. Since probabilities can be neither negative nor greater than 1, we have... 

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