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 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 {\boldsymbol {\varepsilon {ANear}}}} and ε A M e r {\displaystyle {\boldsymbol {\varepsilon {AMer}}}} are shown to be full supercategories of various well-known categories, including the category s T o p {\displaystyle {\boldsymbol {sTop}}} of symmetric topological spaces and continuous maps, and the category M e t ∞ {\displaystyle {\boldsymbol {Met^{\infty }}}} of extended metric spaces and nonexpansive maps. The notation A ↪ B {\displaystyle {\boldsymbol {A}}\hookrightarrow {\boldsymbol {B}}} reads category A {\displaystyle {\boldsymbol {A}}} is embedded in category B {\displaystyle {\boldsymbol {B}}} . The categories ε A M e r {\displaystyle {\boldsymbol {\varepsilon AMer}}} and ε A N e a r {\displaystyle {\boldsymbol {\varepsilon ANear}}} are supercategories for a variety of familiar categories shown in Fig. 3. Let ε A N e a r {\displaystyle {\boldsymbol {\varepsilon {ANear}}}} denote the category of all ε {\displaystyle \varepsilon } -approach nearness spaces and contractions, and let ε A M e r {\displaystyle {\boldsymbol {\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 {\boldsymbol {sTop}}} , the symmetric form of T o p {\displaystyle {\boldsymbol {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 {\boldsymbol {Met^{\infty }}}} with objects that are extended metric spaces is a subcategory of ε A P {\displaystyle {\boldsymbol {\varepsilon AP}}} (having objects ε {\displaystyle \varepsilon } -approach spaces and contractions) (see also). Let ρ X , ρ Y {\displaystyle \rho _{X},\rho _{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,\rho _{X})\longrightarrow (Y,\rho _{Y})} is a contraction if and only if f : ( X , ν D ρ X ) ⟶ ( Y , ν D ρ Y ) {\displaystyle f:(X,\nu _{D_{\rho _{X}}})\longrightarrow (Y,\nu _{D_{\rho _{Y}}})} is a contraction. For nonempty subsets A , B ∈ 2 X {\displaystyle A,B\in 2^{X}} , the distance function D ρ : 2 X × 2 X ⟶ [ 0 , ∞ ] {\displaystyle D_{\rho }:2^{X}\times 2^{X}\longrightarrow [0,\infty ]} is defined by
D ρ ( A , B ) = { inf { ρ ( a , b ) : a ∈ A , b ∈ B } , if A and B are not empty , ∞ , if A or B is empty . {\displaystyle D_{\rho }(A,B)={\begin{cases}\inf {\{\rho (a,b):a\in A,b\in B\}},&{\text{if }}A{\text{ and }}B{\text{ are not empty}},\\\infty ,&{\text{if }}A{\text{ or }}B{\text{ is empty}}.\end{cases}}}
Thus ε {\displaystyle {\boldsymbol {\varepsilon }}} AP is embedded as a full subcategory in ε A N e a r {\displaystyle {\boldsymbol {\varepsilon {ANear}}}} by the functor F : ε A P ⟶ ε A N e a r {\displaystyle F:{\boldsymbol {\varepsilon {AP}}}\longrightarrow {\boldsymbol {\varepsilon {ANear}}}} defined by F ( ( X , ρ ) ) = ( X , ν D ρ ) {\displaystyle F((X,\rho ))=(X,\nu _{D_{\rho }})} and F ( f ) = f {\displaystyle F(f)=f} . Then f : ( X , ρ X ) ⟶ ( Y , ρ Y ) {\displaystyle f:(X,\rho _{X})\longrightarrow (Y,\rho _{Y})} is a contraction if and only if f : ( X , ν D ρ X ) ⟶ ( Y , ν D ρ Y ) {\displaystyle f:(X,\nu _{D_{\rho _{X}}})\longrightarrow (Y,\nu _{D_{\rho _{Y}}})} is a contraction. Thus ε A P {\displaystyle {\boldsymbol {\varepsilon {AP}}}} is embedded as a full subcategory in ε A N e a r {\displaystyle {\boldsymbol {\varepsilon {ANear}}}} by the functor F : ε A P ⟶ ε A N e a r {\displaystyle F:{\boldsymbol {\varepsilon {AP}}}\longrightarrow {\boldsymbol {\varepsilon {ANear}}}} defined by F ( ( X , ρ ) ) = ( X , ν D ρ ) {\displaystyle F((X,\rho ))=(X,\nu _{D_{\rho }})} and F ( f ) = f . {\displaystyle F(f)=f.} Since the category M e t ∞ {\displaystyle {\boldsymbol {Met^{\infty }}}} of extended metric spaces and nonexpansive maps is a full subcategory of ε A P {\displaystyle {\boldsymbol {\varepsilon {AP}}}} , therefore, ε A N e a r {\displaystyle {\boldsymbol {\varepsilon {ANear}}}} is also a full supercategory of M e t ∞ {\displaystyle {\boldsymbol {Met^{\infty }}}} . The category ε A N e a r {\displaystyle {\boldsymbol {\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.
Nearness of sets The adjective near in the context of near sets is used to denote the fact that observed feature value differences of distinct objects are small enough to be considered indistinguishable, i.e., within some tolerance. The exact idea of closeness or 'resemblance' or of 'being within tolerance' is universal enough to appear, quite naturally, in almost any mathematical setting (see, e.g.,). It is especially natural in mathematical applications: practical problems, more often than not, deal with approximate input data and only require viable results with a tolerable level of error. The words near and far are used in daily life and it was an incisive suggestion of F. Riesz that these intuitive concepts be made rigorous. He introduced the concept of nearness of pairs of sets at the ICM in Rome in 1908. This concept is useful in simplifying teaching calculus and advanced calculus. For example, the passage from an intuitive definition of continuity of a function at a point to its rigorous epsilon-delta definition is sometime difficult for teachers to explain and for students to understand. Intuitively, continuity can be explained using nearness language, i.e., a function f : R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} } is continuous at a point c {\displaystyle c} , provided points { x } {\displaystyle \{x\}} near c {\displaystyle c} go into points { f ( x ) } {\displaystyle \{f(x)\}} near f ( c ) {\displaystyle f(c)} . Using Riesz's idea, this definition can be made more precise and its contrapositive is the familiar definition.
Generalization of set intersection From a spatial point of view, nearness (a.k.a. proximity) is considered a generalization of set intersection. For disjoint sets, a form of nearness set intersection is defined in terms of a set of objects (extracted from disjoint sets) that have similar features within some tolerance (see, e.g., §3 in). For example, the ovals in Fig. 1 are considered near each other, since these ovals contain pairs of classes that display similar (visually indistinguishable) colours.
Efremovič proximity space Let X {\displaystyle X} denote a metric topological space that is endowed with one or more proximity relations and let 2 X {\displaystyle 2^{X}} denote the collection of all subsets of X {\displaystyle X} . The collection 2 X {\displaystyle 2^{X}} is called the power set of X {\displaystyle X} . There are many ways to define Efremovič proximities on topological spaces (discrete proximity, standard proximity, metric proximity, Čech proximity, Alexandroff proximity, and Freudenthal proximity), For details, see § 2, pp. 93–94 in. The focus here is on standard proximity on a topological space. For A , B ⊂ X {\displaystyle A,B\subset X} , A {\displaystyle A} is near B {\displaystyle B} (denoted by A δ B {\displaystyle A\ \delta \ B} ), provided their closures share a common point. The closure of a subset A ∈ 2 X {\displaystyle A\in 2^{X}} (denoted by cl ( A ) {\displaystyle {\mbox{cl}}(A)} ) is the usual Kuratowski closure of a set, introduced in § 4, p. 20, is defined by
cl ( A ) = { x ∈ X : D ( x , A ) = 0 } , where D ( x , A ) = inf { d ( x , a ) : a ∈ A } . {\displaystyle {\begin{aligned}{\mbox{cl}}(A)&=\left\{x\in X:D(x,A)=0\right\},\ {\mbox{where}}\\D(x,A)&=\inf \left\{d(x,a):a\in A\right\}.\end{aligned}}}
I.e., cl ( A ) {\displaystyle {\mbox{cl}}(A)} is the set of all points x {\displaystyle x} in X {\displaystyle X} that are close to A {\displaystyle A} ( D ( x , A ) {\displaystyle D(x,A)} is the Hausdorff distance (see § 22, p. 128, in) between x {\displaystyle x} and the set A {\displaystyle A} and d ( x , a ) = | x − a | {\displaystyle d(x,a)=\left|x-a\right|} (standard distance)). A standard proximity relation is defined by
δ = { ( A , B ) ∈ 2 X × 2 X : cl ( A ) ∩ cl ( B ) ≠ ∅ } . {\displaystyle \delta =\left\{(A,B)\in 2^{X}\times 2^{X}:{\mbox{cl}}(A)\ \cap \ {\mbox{cl}}(B)\neq \emptyset \right\}.}
Whenever sets A {\displaystyle A} and B {\displaystyle B} have no points in common, the sets are farfrom each other (denoted A δ _ B {\displaystyle A\ {\underline {\delta }}\ B} ). The following EF-proximity space axioms are given by Jurij Michailov Smirnov based on what Vadim Arsenyevič Efremovič introduced during the first half of the 1930s. Let A , B , E ∈ 2 X {\displaystyle A,B,E\in 2^{X}} .
EF.1 If the set A {\displaystyle A} is close to B {\displaystyle B} , then B {\displaystyle B} is close to A {\displaystyle A} . EF.2
A ∪ B {\displaystyle A\cup B} is close to E {\displaystyle E} , if and only if, at least one of the sets A {\displaystyle A} or B {\displaystyle B} is close to E {\displaystyle E} . EF.3 Two points are close, if and only if, they are the same point. EF.4 All sets are far from the empty set ∅ {\displaystyle \emptyset } . EF.5 For any two sets A {\displaystyle A} and B {\displaystyle B} which are far from each other, there exists C , D ∈ 2 X {\displaystyle C,D\in 2^{X}} , C ∪ D = X {\displaystyle C\cup D=X} , such that A {\displaystyle A} is far from C {\displaystyle C} and B {\displaystyle B} is far from D {\displaystyle D} (Efremovič-axiom). The pair ( X , δ ) {\displaystyle (X,\delta )} is called an EF-proximity space. In this context, a space is a set with some added structure. With a proximity space X {\displaystyle X} , the structure of X {\displaystyle X} is induced by the EF-proximity relation δ {\displaystyle \delta } . In a proximity space X {\displaystyle X} , the closure of A {\displaystyle A} in X {\displaystyle X} coincides with the intersection of all closed sets that contain A {\displaystyle A} .
Theorem 1 The closure of any set A {\displaystyle A} in the proximity space X {\displaystyle X} is the set of points x ∈ X {\displaystyle x\in X} that are close to A {\displaystyle A} .
Visualization of EF-axiom
Let the set X {\displaystyle X} be represented by the points inside the rectangular region in Fig. 5. Also, let A , B {\displaystyle A,B} be any two non-intersection subsets (i.e. subsets spatially far from each other) in X {\displaystyle X} , as shown in Fig. 5. Let C c = X ∖ C {\displaystyle C^{c}=X\backslash C} (complement of the set C {\displaystyle C} ). Then from the EF-axiom, observe the following:
A
δ _ B , B ⊂ C , D = C c , X = D ∪ C , A ⊂ D , hence, we can write A δ _ B ⇒ A δ _ C and B δ _ D , for some C , D in X so that C ∪ D = X . ◼ {\displaystyle {\begin{aligned}A&{}\mathrel {\underline {\delta }} B,\\B&\subset C,\\D&=C^{c},\\X&=D\cup C,\\A&\subset D,\ {\mbox{hence, we can write}}\\A\ {\underline {\delta }}\ B\ &\Rightarrow \ A\ {\underline {\delta }}\ C\ {\mbox{and}}\ B\ {\underline {\delta }}\ D,\ {\mbox{for some}}\ C,D\ {\mbox{in}}\ X{\mbox{ so that }}C\cup D=X.\qquad \blacksquare \end{aligned}}}
Descriptive proximity space Descriptively near sets were introduced as a means of solving classification and pattern recognition problems arising from disjoint sets that resemble each other. Recently, the connections between near sets in EF-spaces and near sets in descriptive EF-proximity spaces have been explored in. Again, let X {\displaystyle X} be a metric topological space and let Φ = { ϕ 1 , … , ϕ n } {\displaystyle \Phi =\left\{\phi _{1},\dots ,\phi _{n}\right\}} a set of probe functions that represent features of each x ∈ X {\displaystyle x\in X} . The assumption made here is X {\displaystyle X} contains non-abstract points that have measurable features such as gradient orientation. A non-abstract point has a location and features that can be measured (see § 3 in ). A probe function ϕ : X → R {\displaystyle \phi :X\rightarrow \mathbb {R} } represents a feature of a sample point in X {\displaystyle X} . The mapping Φ : X ⟶ R n {\displaystyle \Phi :X\longrightarrow \mathbb {R} ^{n}} is defined by Φ ( x ) = ( ϕ 1 ( x ) , … , ϕ n ( x ) ) {\displaystyle \Phi (x)=(\phi _{1}(x),\dots ,\phi _{n}(x))} , where R n {\displaystyle \mathbb {R} ^{n}} is an n-dimensional real Euclidean vector space. Φ ( x ) {\displaystyle \Phi (x)} is a feature vector for x {\displaystyle x} , which provides a description of x ∈ X {\displaystyle x\in X} . For example, this leads to a proximal view of sets of picture points in digital images. To obtain a descriptive proximity relation (denoted by δ Φ {\displaystyle \delta _{\Phi }} ), one first chooses a set of probe functions. Let Q : 2 X ⟶ 2 R n {\displaystyle {\mathcal {Q}}:2^{X}\longrightarrow 2^{R^{n}}} be a mapping on a subset of 2 X {\displaystyle 2^{X}} into a subset of 2 R n {\displaystyle 2^{R^{n}}} . For example, let A , B ∈ 2 X {\displaystyle A,B\in 2^{X}} and Q ( A ) , Q ( B ) {\displaystyle {\mathcal {Q}}(A),{\mathcal {Q}}(B)} denote sets of descriptions of points in A , B {\displaystyle A,B} , respectively. That is,
Q ( A ) = { Φ ( a ) : a ∈ A } , Q ( B ) = { Φ ( b ) : b ∈ B } . {\displaystyle {\begin{aligned}{\mathcal {Q}}(A)&=\left\{\Phi (a):a\in A\right\},\\{\mathcal {Q}}(B)&=\left\{\Phi (b):b\in B\right\}.\end{aligned}}}
The expression A δ Φ B {\displaystyle A\mathrel {\delta _{\Phi }} B} reads A {\displaystyle A} is descriptively near B {\displaystyle B} . Similarly, A δ _ Φ B {\displaystyle A\mathrel {{\underline {\delta }}_{\Phi }} B} reads A {\displaystyle A} is descriptively far from B {\displaystyle B} . The descriptive proximity of A {\displaystyle A} and B {\displaystyle B} is defined by
A δ Φ B ⇔ Q ( cl ( A ) ) δ Q ( cl ( B ) ) ≠ ∅ . {\displaystyle A\mathrel {\delta _{\Phi }} B\Leftrightarrow {\mathcal {Q}}({\mbox{cl}}(A))\mathrel {\delta } {\mathcal {Q}}({\mbox{cl}}(B))\neq \emptyset .}
The descriptive intersection ∩ Φ {\displaystyle \