Metric topology

In order to define the metric topology, we must first introduce the notion of metric, a function cl AxA->7 isa metric on a set A if and only if it satisfies... 

A fundamental theorem states that a function / 7i -> 72 between two metric topologies is continuous if and only if for all open sets C/ 72, the set f U) is open in 72- In particular, if two different metrics, d and d, give rise to the same family of open sets then any function which is continuous under d will also be continuous under 82. 

If the topology T is chosen as the metric topology, that is, if the T-open sets are precisely those which are open in some metric d introduced into the set X, then one obtains the metric topological space (X, T). Note that the metric topological space (X, T) is a Hausdorff space and also a normal space. 

Three groups of invariants presented in Appendixes are considered metric, topological and information indices. Metric indices of graphs have been studied in many papers of graph theory. A bibliography on the subjects is given in [57, 65]. Topological indices are well-known in... 

Take three fuzzy sets A, B, and C and their a-cuts G (a), Gg(a), and G ia), respectively, for each membership function value a. Assume that the a-cuts GJ.a), Gg(a and G(-(a) depend at least piecewise continuously on the a parameter from the unit interval [0,1], where the intervals of continuity have nonzero lengths and where continuity is understood within the metric topology of the underlying space X. For the three pairs formed from these three fuzzy sets, the ordinary Hausdorff distances h(G (a),Gg(a)), h(Gg(a), Gcia)), and h GJ,a ... 

To the present authors best knowledge, the term chemical topology originates with V. Prelog who introduced it for certain model concepts of the stereochemistry of individual molecules (see ReL and and Ref. cited therein). The concept of chemical topology as a metric topology whose subsets represent the constitutional aspect of chemical systems and their relations, however, seems to be novel. 

The quantity df defined by this formula is known as the dimensionality of the object. More precisely, it is the so-called fractal, scaling, or Hausdorff dimensionality. (In maths, you may hear of lots of others, e.g. metric, topological, etc., but we shall not talk about them.)... 

The norm defines the metric topology from U, which is interpreted as uniform topology, according to which the neighborhood of an element A U is given as follows ... 

Calculus computer science geometry measure theory metric topology probability set theory. 

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... 

As should have been made plausible by the above discussion, symbolic dynamics provides an intuitive conceptual bridge between continuous and discrete dynamical systems. On the one hand, except for the fact that the coarse-grained discrete dynamics of symbolic systems is typically nonlocal in character (see the following section), symbolic dynamical systems are essentially CA-in-disguise. On the other hand, by exploiting the fundamental CA property of continuity in the Cantor-set topology under the metric defined by equation 4.49, real-valued representations of CA dynamics may be readily obtained. We discuss these two alternative points of... 

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... 

The procedure of DG calculations can be subdivided in three separated steps [119-121]. At first, holonomic matrices (see below for explanahon) with pairwise distance upper and lower limits are generated from the topology of the molecule of interest. These limits can be further restrained by NOE-derived distance information which are obtained from NMR experiments. In a second step, random distances within the upper and lower limit are selected and are stored in a metric matrix. This operation is called metrization. Finally, all distances are converted into a complex geometry by mathematical operations. Hereby, the matrix-based distance space is projected into a Gartesian coordinate space (embedding). 

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