Near sets

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. 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 nearnessbetween 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. 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 δ {displaystyle delta } -maps), Unif (uniform spaces and uniformly continuous maps) and Cont (contiguity spaces and contiguity maps) as embedded full subcategories. The categories ε A N e a r {displaystyle {oldsymbol {varepsilon {ANear}}}} and ε A M e r {displaystyle {oldsymbol {varepsilon {AMer}}}} are shown to be full supercategories of various well-known categories, including the category s T o p {displaystyle {oldsymbol {sTop}}} of symmetric topological spaces and continuous maps, and the category M e t ∞ {displaystyle {oldsymbol {Met^{infty }}}} of extended metric spaces and nonexpansive maps. The notation A ↪ B {displaystyle {oldsymbol {A}}hookrightarrow {oldsymbol {B}}} reads category A {displaystyle {oldsymbol {A}}} is embedded in category B {displaystyle {oldsymbol {B}}} . The categories ε A M e r {displaystyle {oldsymbol {varepsilon AMer}}} and ε A N e a r {displaystyle {oldsymbol {varepsilon ANear}}} are supercategories for a variety of familiar categories shown in Fig. 3. Let ε A N e a r {displaystyle {oldsymbol {varepsilon {ANear}}}} denote the category of all ε {displaystyle varepsilon } -approach nearness spaces and contractions, and let ε A M e r {displaystyle {oldsymbol {varepsilon AMer}}} denote the category of all ε {displaystyle varepsilon } -approach merotopic spaces and contractions. Among these familiar categories is s T o p {displaystyle {oldsymbol {sTop}}} , the symmetric form of T o p {displaystyle {oldsymbol {Top}}} (see category of topological spaces), the category with objects that are topological spaces and morphisms that are continuous maps between them. M e t ∞ {displaystyle {oldsymbol {Met^{infty }}}} with objects that are extended metric spaces is a subcategory of ε A P {displaystyle {oldsymbol {varepsilon AP}}} (having objects ε {displaystyle varepsilon } -approach spaces and contractions) (see also). Let ρ X , ρ Y {displaystyle ho _{X}, ho _{Y}} be extended pseudometrics on nonempty sets X , Y {displaystyle X,Y} , respectively. The map f : ( X , ρ X ) ⟶ ( Y , ρ Y ) {displaystyle f:(X, ho _{X})longrightarrow (Y, ho _{Y})} is a contraction if and only if f : ( X , ν D ρ X ) ⟶ ( Y , ν D ρ Y ) {displaystyle f:(X, u _{D_{ ho _{X}}})longrightarrow (Y, u _{D_{ ho _{Y}}})} is a contraction. For nonempty subsets A , B ∈ 2 X {displaystyle A,Bin 2^{X}} , the distance function D ρ : 2 X × 2 X ⟶ [ 0 , ∞ ] {displaystyle D_{ ho }:2^{X} imes 2^{X}longrightarrow } is defined by Thus ε {displaystyle {oldsymbol {varepsilon }}} AP is embedded as a full subcategory in ε A N e a r {displaystyle {oldsymbol {varepsilon {ANear}}}} by the functor F : ε A P ⟶ ε A N e a r {displaystyle F:{oldsymbol {varepsilon {AP}}}longrightarrow {oldsymbol {varepsilon {ANear}}}} defined by F ( ( X , ρ ) ) = ( X , ν D ρ ) {displaystyle F((X, ho ))=(X, u _{D_{ ho }})} and F ( f ) = f {displaystyle F(f)=f} . Then f : ( X , ρ X ) ⟶ ( Y , ρ Y ) {displaystyle f:(X, ho _{X})longrightarrow (Y, ho _{Y})} is a contraction if and only if f : ( X , ν D ρ X ) ⟶ ( Y , ν D ρ Y ) {displaystyle f:(X, u _{D_{ ho _{X}}})longrightarrow (Y, u _{D_{ ho _{Y}}})} is a contraction. Thus ε A P {displaystyle {oldsymbol {varepsilon {AP}}}} is embedded as a full subcategory in ε A N e a r {displaystyle {oldsymbol {varepsilon {ANear}}}} by the functor F : ε A P ⟶ ε A N e a r {displaystyle F:{oldsymbol {varepsilon {AP}}}longrightarrow {oldsymbol {varepsilon {ANear}}}} defined by F ( ( X , ρ ) ) = ( X , ν D ρ ) {displaystyle F((X, ho ))=(X, u _{D_{ ho }})} and F ( f ) = f . {displaystyle F(f)=f.} Since the category M e t ∞ {displaystyle {oldsymbol {Met^{infty }}}} of extended metric spaces and nonexpansive maps is a full subcategory of ε A P {displaystyle {oldsymbol {varepsilon {AP}}}} , therefore, ε A N e a r {displaystyle {oldsymbol {varepsilon {ANear}}}} is also a full supercategory of M e t ∞ {displaystyle {oldsymbol {Met^{infty }}}} . The category ε A N e a r {displaystyle {oldsymbol {varepsilon {ANear}}}} is a topological construct. The notions of near and far in mathematics can be traced back to works by Johann Benedict Listing and Felix Hausdorff. The related notions of resemblance and similarity can be traced back to J.H. Poincaré, who introduced sets of similar sensations (nascent tolerance classes) to represent the results of G.T. Fechner's sensation sensitivity experiments and a framework for the study of resemblance in representative spaces as models of what he termed physical continua. The elements of a physical continuum (pc) are sets of sensations. The notion of a pc and various representative spaces (tactile, visual, motor spaces) were introduced by Poincaré in an 1894 article on the mathematical continuum, an 1895 article on space and geometry and a compendious 1902 book on science and hypothesis followed by a number of elaborations, e.g.,. The 1893 and 1895 articles on continua (Pt. 1, ch. II) as well as representative spaces and geometry (Pt. 2, ch IV) are included as chapters in. Later, F. Riesz introduced the concept of proximity or nearness of pairs of sets at the International Congress of Mathematicians (ICM) in 1908. During the 1960s, E.C. Zeeman introduced tolerance spaces in modelling visual perception. A.B. Sossinsky observed in 1986 that the main idea underlying tolerance space theory comes from Poincaré, especially. In 2002, Z. Pawlak and J. Peters considered an informal approach to the perception of the nearness of physical objects such as snowflakes that was not limited to spatial nearness. In 2006, a formal approach to the descriptive nearness of objects was considered by J. Peters, A. Skowron and J. Stepaniuk in the context of proximity spaces. In 2007, descriptively near sets were introduced by J. Peters followed by the introduction of tolerance near sets. Recently, the study of descriptively near sets has led to algebraic, topological and proximity space foundations of such sets.