Abstract
Integer and digital spaces are playing a significant role in digital image processing, computer graphics, computer tomography, robot vision, and many other fields dealing with finitely or countable many objects. It is proven here that every finite T0-space is a quotient space of a subspace of some simplex, i.e. of some subspace of a Euclidean space. Thus finite and digital spaces can be considered as abstract simplicial structures of subspaces of Euclidean spaces. Primitive subspaces of finite, digital, and integer spaces are introduced. They prove to be useful in the investigation of connectedness structure, which can be represented as a poset, and also in consideration of the dimension of finite spaces. Essentially T0-spaces and finitely connected and primitively path connected spaces are discussed.
Highlights
Large-scale systems in fields such as computing, economics, ecology, and engineering are becoming more complex
See references [7,8,9] for necessary definitions and terms such as: are, curve, integer and digital lines, integer spaces, path eonnectedness, and digital topology, as well as for results which are essential for following the arguments of this paper
Familiarity with topology discussed in references [1,2,3,4] is assumed
Summary
Integer and digital spaces are playing a significant role in digital image processing, computer graphics, computer tomography, robot vision, and many other fields dealing with finitely or countable many objects. F’mite and digital spaces can be considered as abstract simplicial structures of subspaces of Euclidean spaces. They prove to be useful in the investigation of connectedness structure, which can be represented as a poset, and in consideration of the dimension of finite spaces. T0-spaces and finitely connected and primitively path connected spaces are discussed. This paper continues to build the theory of finite, digital, and integer spaces. See references [7,8,9] for necessary definitions and terms such as: are, curve, integer and digital lines, integer spaces, path eonnectedness, and digital topology, as well as for results which are essential for following the arguments of this paper. Familiarity with topology discussed in references [1,2,3,4] is assumed
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