Point-set topology

D. L Stand and M. L. Stand, Real Analysis with Point-Set Topology (1987)... 

A technical point How we can be sure that there actually is a limiting set We invoke a standard theorem from point-set topology. Observe that the successive images of the square are nested inside each other like Chinese boxes ... 

Point-Set Topology. The study of sets starts from the notion of a set as a collection of elements. In the simplest cases, one can think of this collection as a... 

The interest of point-set topology outside the area of set theory comes from considering what happens when functions are applied to various kinds of sets of points. In particular, one is frequently interested in continuous functions, which do not allow for jumps in the values of the function. Traditionally, a physical model for a function suggested that it ought to be continuous, but the laws of quantum mechanics have made the assumption of continuity a little less usefiil and have prodnced the notion of a distribution as a more general sort of function than the continuous case. 

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... 

Brillouin Leon (1889-1969) French born US phys., invented the concept of negentropy, father of information theory (book Theorie de 1 Information 1959) Brouwer Luitzen Egbertus Jan (1881-1966) Dutch, math., founder of modern topology, proved that dimensionality of a Cartesian space is topological invariant, worked in point sets... 

The Kohonen network or self-organizing map (SOM) was developed by Teuvo Kohonen [11]. It can be used to classify a set of input vectors according to their similarity. The result of such a network is usually a two-dimensional map. Thus, the Kohonen network is a method for projecting objects from a multidimensional space into a two-dimensional space. This projection keeps the topology of the multidimensional space, i.e., points which are close to one another in the multidimensional space are neighbors in the two-dimensional space as well. An advantage of this method is that the results of such a mapping can easily be visualized. 

That Dfractai givcs the expected result for simple sets in Euclidean space is easy to see. If A consists of a single point, for example, we have N A, e) = 1, Ve, and thus that / fractal = 0. Similarly, if A is a line segment of length L, then N A,e) L/e so that Dfraciai = 1. In fact, for the usual n dinien.sional Euclidean sets, the fractal dimension equals the topological dimension. There are nrore complicated sets, however, for wliic h the two measures differ. 

Consider a set X. The topological space (X, T) is called a Hausdorff space if for any two distinct points x, y eX there exist disjoint T-open sets Tx, Ty,... 

Consider an K-dimensional set X. Set X is simply connected if and only if every -dimensional (k < n) topological sphere Sk in set X is contractible to a point. 

The mathematical abstraction of the topology of a pipeline network is called a graph which consists of a set of vertices (sometimes also referred to as nodes, junctions, or points)... 

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