Dimension for Alexandrov spaces

Wiederhold, Petra; Wilson, Richard G.

doi:10.1117/12.142181

Cited by 11 publications

(9 citation statements)

References 6 publications

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“…Essentially the same concepts A dimo (p), A dim(G), T dim G (p) and T dim(G) were defined for Alexandroff spaces by Wiederhold and Wilson in [19], where they were called ODIL p, ODIM G, DIL p and DIM G respectively.…”

Section: Three Definitions Of Dimensionmentioning

confidence: 99%

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Dimensional properties of graphs and digital spaces

Evako¹,

Kopperman

Mukhin

1996

J Math Imaging Vis

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Many applications of digital image processing now deal with three-and higher-dimensional images.One way to represent n-dimensional digital images is to use the specialization graphs of subspaces of the Alexandroff topological space Z n (where Z denotes the integers with the Khalimsky line topology). In this paper the dimension of any such graph is defined in three ways, and the equivalence of the three definitions is established. Two of the definitions have a geometric basis and are closely related to the topological definition of inductive dimension; the third extends the Alexandroff dimension to graphs. Diagrams are given of graphs that are dimensionally correct discrete models of Euclidean spaces, n-dimensional spheres, a projective plane and a torus. New characterizations of n-dimensional (digital) surfaces are presented. Finally, the local structure of the space Z n is analyzed, and it is shown that Z n is an n-dimensional surface for all n > 1.

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Section: Three Definitions Of Dimensionmentioning

confidence: 99%

“…[] The result that T dim(g) = A dim(g) for all partially ordered sets G is not new--see Proposition 6 in [19].…”

Section: Theorem 3 If G Is Transitive Then a Dim(g) = I Dim(g) And mentioning

confidence: 99%

Dimensional properties of graphs and digital spaces

Evako¹,

Kopperman

Mukhin

1996

J Math Imaging Vis

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“…Then for ind we suggest decomposition, sum and product theorems in the class (Propositions 2.2, 2.3 and 2.5, respectively). Let us note that the product theorem is written as an equality and thus it is stronger than the theorem from [WW1]. The sum and product theorems there we prove even for the small transfinite inductive dimension trind (Propositions 4.3 and 4.4).…”

Section: Introductionmentioning

confidence: 76%

“…Remark 2.6. Let us notice that the inequality ind (X × Y) ≤ ind X + ind Y for non-empty Alexandroff T 0 -spaces X, Y was announced in [WW1].…”

Section: Properties Of the Small Inductive Dimension In Alexandroff Spacesmentioning

confidence: 99%

The small inductive dimension of subsets of Alexandroff spaces

Chatyrko

Han

Hattori

2016

Filomat

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We describe the small inductive dimension ind in the class of Alexandroff spaces by the use of some standard spaces. Then for ind we suggest decomposition, sum and product theorems in the class. The sum and product theorems there we prove even for the small transfinite inductive dimension trind. As an application of that, for each positive integers k,n such that k ? n we get a simple description in terms of even and odd numbers of the family S(k,n) = {S ? Kn : |S|=k+1 and indS=k}, where K is the Khalimsky line.

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“…Hence, the dimension of a cell is its depth in the "crater" of interlocked neighborhoods. This definition is equivalent to the known notion of the dimension or height of an element of a partially ordered set (poset) as defined in the theory of poset topology [3] and of A-spaces [26].…”

Section: A Relation Between Ssts and Khalimsky Spacesmentioning

confidence: 99%

Properties of space set topological spaces

Han

2016

Filomat

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Since a locally finite topological structure plays an important role in the fields of pure and applied topology, the paper studies a special kind of locally finite spaces, so called a space set topology (for brevity, SST) and further, proves that an SST is an Alexandroff space satisfying the separation axiom T0. Unlike a point set topology, since each element of an SST is a space, the present paper names the topology by the space set topology. Besides, for a connected topological space (X,T) with |X| = 2 the axioms T0, semi-T1/2 and T1/2 are proved to be equivalent to each other. Furthermore, the paper shows that an SST can be used for studying both continuous and digital spaces so that it plays a crucial role in both classical and digital topology, combinatorial, discrete and computational geometry. In addition, a connected SST can be a good example showing that the separation axiom semi-T1/2 does not imply T1/2.

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Dimension for Alexandrov spaces

Cited by 11 publications

References 6 publications

Dimensional properties of graphs and digital spaces

Dimensional properties of graphs and digital spaces

The small inductive dimension of subsets of Alexandroff spaces

Properties of space set topological spaces

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