In mathematics, near sets are either spatially close or descriptively close. Spatially close sets have nonempty intersection. In other words, spatially close sets are not disjoint sets, since they always have at least one element in common. Descriptively close sets contain elements that have matching descriptions. Such sets can be either disjoint or non-disjoint sets. Spatially near sets are also descriptively near sets.
The underlying assumption with descriptively close sets is that such sets contain elements that have location and measurable features such as colour and frequency of occurrence. The description of the element of a set is defined by a feature vector. Comparison of feature vectors provides a basis for measuring the closeness of descriptively near sets. Near set theory provides a formal basis for the observation, comparison, and classification of elements in sets based on their closeness, either spatially or descriptively. Near sets offer a framework for solving problems based on human perception that arise in areas such as image processing, computer vision as well as engineering and science problems.
Near sets have a variety of applications in areas such as topology, pattern detection and classification, abstract algebra, mathematics in computer science, and solving a variety of problems based on human perception that arise in areas such as image analysis, image processing, face recognition, ethology, as well as engineering and science problems. From the beginning, descriptively near sets have proved to be useful in applications of topology, and visual pattern recognition , spanning a broad spectrum of applications that include camouflage detection, micropaleontology, handwriting forgery detection, biomedical image analysis, content-based image retrieval, population dynamics, quotient topology, textile design, visual merchandising, and topological psychology.
As an illustration of the degree of descriptive nearness between two sets, consider an example of the Henry colour model for varying degrees of nearness between sets of picture elements in pictures (see, e.g., §4.3). The two pairs of ovals in Fig. 1 and Fig. 2 contain coloured segments. Each segment in the figures corresponds to an equivalence class where all pixels in the class have similar descriptions, i.e., picture elements with similar colours. The ovals in Fig.1 are closer to each other descriptively than the ovals in Fig. 2.
History
It has been observed that the simple concept of nearness unifies various concepts of topological structures inasmuch as the category Near of all nearness spaces and nearness preserving maps contains categories sTop (symmetric topological spaces and continuous maps), Prox (proximity spaces and -maps), Unif (uniform spaces and uniformly continuous maps) and Cont (contiguity spaces and contiguity maps) as embedded full subcategories. The categories and are shown to be full supercategories of various well-known categories, including the category of symmetric topological spaces and continuous maps, and the category of extended metric spaces and nonexpansive maps. The notation reads category is embedded in category . The categories and are supercategories for a variety of familiar categories shown in Fig. 3. Let denote the category of all -approach nearness spaces and contractions, and let denote the category of all -approach merotopic spaces and contractions.
Among these familiar categories is , the symmetric form of (see category of topological spaces), the category with objects that are topological spaces and morphisms that are continuous maps between them. with objects that are extended metric spaces is a subcategory of (having objects -approach spaces and contractions) (see also). Let be extended pseudometrics on nonempty sets , respectively. The map is a contraction if and only if is a contraction. For nonempty subsets , the distance function is defined by
Thus AP is embedded as a full subcategory in by the functor defined by and . Then is a contraction if and only if is a contraction. Thus is embedded as a full subcategory in by the functor defined by and Since the category of extended metric spaces and nonexpansive maps is a full subcategory of , therefore, is also a full supercategory of . The category is a topological construct.