Topological Spaces By Eduard Cech. Revised edition by Z. Frolik and M. Katetov. Pp. 893. (Prague: Publishing House of the Czechoslovak Academy of Sciences; London and New York: Interscience Publishers, a Division of John Wiley and Sons, 1966.) 145s.

Point Set Topology

The main task of digital image processing is to recognize properties of real objects based on their digital images. These images are obtained by some sampling device, like a CCD camera, and represented as finite sets of points that are assigned some value in a gray-level or color scale. Based on technical properties of sampling devices, these points are usually assumed to form a square grid and are modeled as finite subsets of Z^2. Therefore, a fundamental question in digital image processing is which features in the digital image correspond, under certain conditions, to properties of the underlying objects. In practical applications this question is mostly answered by visually judging the obtained digital images. In this paper we present a comprehensive answer to this question with respect to topological properties. In particular, we derive conditions relating properties of real objects to the grid size of the sampling device which guarantee that a real object and its digital image are topologically equivalent. These conditions also imply that two digital images of a given object are topologically equivalent. This means, for example, that shifting or rotating an object or the camera cannot lead to topologically different images, i.e., topological properties of obtained digital images are invariant under shifting and rotation.

https://www.cis.temple.edu/~latecki/Papers/jmiv98.pdf

Preserving Topology by a Digitization Process

McKinsey J. C. C. and Tarski Alfred. On closed elements in closure algebras. Annals of mathematics , ser. 2 vol. 47 (1946), pp. 122–162.

Semi-proximity continuous functions in digital images

The present paper is devoted to the study of grill determined L-approachmerotopological spaces. The category having such spaces as objects is shown to be a topological construct (its initial and final structures are provided explicitly). The lattice structure of the family of all these spaces is also discussed. In the classical theory, this category (that is, when L={0, 1}) is a supercategory of the category of pseudo metric spaces and nonexpansive maps.

Grill Determined L-Approach Merotopological Spaces

In this paper neighbourhood lattices are developed as a generalisation of topological spaces in order to examine to what extent the concepts of “openness”, “closedness”, and “continuity” defined in topological spaces depend on the lattice structure of P(X) , the power set of X . A general pre-neighbourhood system, which satisfies the poset analogues of the neighbourhood system of points in a topological space, is defined on an ∧-semi-lattice, and is used to define open elements. Neighbourhood systems, which satisfy the poset analogues of the neighbourhood system of sets in a topological space, are introduced and it is shown that it is the conditionally complete atomistic structure of P(X) which determines the extension of pre-neighbourhoods of points to the neighbourhoods of sets. The duals of pre-neighbourhood systems are used to generate closed elements in an arbitrary lattice, independently of closure operators or complementation. These dual systems then form the backdrop for a brief discussion of the relationship between preneighbourhood systems, topological closure operators, algebraic closure operators, and Cech closure operators. Continuity is defined for functions between neighbourhood lattices, and it is proved that a function f: X → Y between topological spaces is continuous if and only if corresponding direct image function between the neighbourhood lattices P(X) and P(Y) is continuous in the neighbourhood sense. Further, it is shown that the algebraic character of continuity, that is, the non-convergence aspects, depends only on the properites of pre-neighbourhood systems. This observation leads to a discussion of the continuity properties of residuated mappings. Finally, the topological properties of normality and regularity are characterised in terms of the continuity properties of the closure operator on a topological space.

/pdf/neighbourhood-lattices-a-poset-approach-to-topological-2q0pj3y35f.pdf

Neighbourhood lattices – a poset approach to topological spaces

The questions which motivated the development of neighborhood (nbhd) lattices as a generalization of topological (top) spaces are discussed. Nbhd systems, which are shown to be appropriate for characterizing continuity, are defined on ^-semilattices, and are used to define open elements. The duals of nbhd systems are used to define closed elements in a lattice, independently of closure operators or complementation. In addition, the top continuity of a function f: XY is characterized in terms of the nbhd continuity of the direct image function mapping P(X), the power set of X, into P(Y). T1-nbhd lattices are defined, independently of points. Finally, the relationship between continuity and convergence is established by proving that a residuated function between conditionally complete T1-nbhd lattices is continuous iff it preserves the limit of convergent nets.

A Step Beyond Topology

Starting with the intuitive concept of `nearness' as a binary relation, semi-proximity spaces (sp-spaces) are defined. The restrictions on semi-proximity spaces are weaker than the restrictions on topological proximity spaces. Nevertheless, semi-proximity spaces generalize classical topological spaces. Moreover, it is possible to describe all digital pictures used in computer vision and computer graphics as non-trivial semi-proximity spaces, which is not possible in classical topology. Therefore, we use semi-proximity spaces to establish a formal relationship between the `topological' concepts of digital image processing and their continuous counterparts in Rn. Especially interesting are continuous functions in semi- proximity spaces. The definition of a `nearly' bicontinuous function is given which does not require the function to be one-to-one. A nearly bicontinuous function preserves connectedness in both directions. Therefore, nearly bicontinuous functions can be used for characterizing well-behaved operations on digital images such as thinning. Further, it is shown that the deletion of a simple point can be treated as a nearly bicontinuous function. These properties and the fact that a variety of nearness relations can be defined on digital pictures indicate that nearly continuous functions are a useful tool in the difficult task of shape description.© (1995) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Nearly continuous functions in digital images

A consideration of the separation properties of pre-neighbourhood lattices, leads to the definition and characterisation of T 1 -neighbourhood lattices in terms of the properties of the neighbourhood mapping, independently of points. It is then shown that if net convergence is defined in neighbourhood lattices as a consequence of replacing ‘point’ by ‘set’ in topological convergence, then the limits of convergent nets are unique. The relationship between continuity and convergence is established with the proof of the statement that a residuated function between conditionally complete T 1 -neighbourhood lattices is continuous if and only if it preserves the limit of convergent nets. If P(X) denotes the power set of X , then the observation that a filter in a topological space ( X, T ) is a net in P(X) leads to a discussion of the net convergence of a filter as a special case of net convergence. Particular attention is paid to maximal filters, Frechet filters and to the filter generated by the limit elements of a net. Further, if the ‘filter’ convergence of a filter F in a topological space ( X, T ) is given by , if η( x ) ⊆ F , then the relationship between ‘filter’ convergence and the net convergence of a filter in P ( X ) is established. Finally, it is proved that, in the neighbourhood system ‘lifted’ from a topological space to P(P(X)) , the continuous image of a filter which converges to a singleton set is a convergent filter with the appropriate image set as the limit.

/pdf/convergence-in-neighbourhood-lattices-2fjelox69q.pdf

Convergence in neighbourhood lattices

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