Bol. Soc. Paran. Mat. (3s.) v. 35 1 (2017): 97–109. c SPM –ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v35i1.24592 On the quotients of c-spaces Santhosh P.K. abstract: In this paper, properties of quotient maps in c-spaces are studied in detail. A method of finding quotient space of topologizable and graphical c-spaces are described. Key Words: c-space,connectivity space, c-continuous function, quotient space, topologizable and graphical c-spaces Contents 1 Introduction 97 2 Preliminaries 98 3 Weak c-structure Generated by a Family of Functions 100 3.1 Examples of Quotient Spaces ......................101 4 Properties of Quotient Spaces 101 5 Quotient Space Verses Equivalence Classes 104 6 Quotient Space of Topologizable c-spaces 104 7 Quotient space of Graphical c-spaces 107 1. Introduction The concept of connected sets can be found in topology, graph theory, order structure and in their fuzzy analogues. A subset A of a topological space X is said to be connected if it cannot be separated into two disjoint non empty open subsets. In this definition, separation is emphasized. When working with graphs, connectivity is defined in terms of paths.A graph G is said to be connected if for every vertices x and y there is a path in G with end vertices x and y. Although these classical definitions have been extensively applied in image pro- cessing and analysis, these two approaches( i.e., separation by open sets and joining points by path) are incompatible [9]. For the sake of completeness of this paper, let us illustrate this with an example [9] here. Consider the digital space Z2. Define two adjacency relations, the 4- adjacency and 8-adjacency, as follows. Let (i, j) and (i′, j′) ∈ Z2. 2000 Mathematics Subject Classification: 54A05, 05C40 BSP Typeset by Mstyle. 97 c Soc. Paran. de Mat. 98 Santhosh P.K. (i, j) is 4-adjacent to (i′, j′) if |i − i′| + |j − j′| =1 and (i, j) is 8-adjacent to (i′, j′) if max{|i − i′|, |j − j′|} =1. Thus we get two graphs with Z2 as set of vertices and edges linking pairs of adjacent points (4 or 8-adjacency respectively). This leads to the classical notion of 4-connectivity and 8-connectivity of subsets of Z2. But it is proved that there is a topology on Z2 for which connected sets are precisely 4-connected sets and there is no topology on Z2 for which connectedness is equivalent to 8-connectedness. Further, it is proved that [2], any odd cycle Cn, n ≥ 5 has no compatible topology. Thus there are graphs where connectivity does not arise from a topology. Conversely there are topological spaces which are not graphical. For example, in [9] it is noted that, in Rn with n ≥ 1, every pair of points is disconnected and that it is not graphical. That is, topological connectivity of Rn cannot arise from a graph. In general, topological connectivity is useful for images defined over a contin- uous space where as graph theoretic connectivity is useful for images defined over a discrete space. Compatibility is essential as discrete images are often obtained from the discretization of the continuous scenes. Thus topological or graph theo- retical approach to connectivity limits the application of connectivity or the use of connected sets in practical purposes. In 1983,Bo¨rger R. [1] proposed an axiomatic approach to connectivity, known as the theory of connectivity classes or c-structures, which overcomes the shortfalls of the classical definitions of connectivity. Here the axioms involve the minimum properties that connected sets can have. That is, one point sets are connected and that union of connected sets having a common point is connected. A systematic study of this space was further carried out by J. Serra [12], H. J. A. M. Hejimans [4] and C. Ronse [9] etc. This space found profound applications in the areas of Image segmentation, Image Filtering, Image coding, Digital Topology, Pattern Recognition, Mathematical Morphology etc [4,5,10,12,13]. In this paper, some ex- isting concepts are formalized and explored some properties of quotient spaces, which paves the foundation for our further work. 2. Preliminaries Let X be a non empty set and CX be a collection of subsets of X such that the following properties hold. (i) φ ∈ CX and {x} ∈ CX for every x ∈ X. (ii) If {Ci : i ∈ I} be a non empty collection of subsets in C with ∩ Ci 6= φ, then i∈I ∪ Ci ∈ CX . i∈I Then CX is called a c-structure on X and X with a c-structure on it is called a c-space. A c-space X is said to be a connective space [7] if CX satisfies two more conditions as given below. On the quotients of c-spaces 99 (iii) Given any non empty sets A, B ∈ CX with A ∪ B ∈ CX , then there exists x ∈ A ∪ B such that {x} ∪ A ∈ CX and {x} ∪ B ∈ CX . (iv) If A, B, Ci ∈ CX are disjoint and A ∪ B ∪ Ci ∈ CX , then there exists J ⊆ I iS∈I such that A ∪ Cj ∈ CX and B ∪ Ci ∈ CX . jS∈J i∈SI−J A c-structure satisfies above two conditions are called a connective structure or connectology on X. Other terminologies used for c-spaces are connectivity space [1,12] and integral connectivity space [3]. In his paper [1], Reinhard Bo¨rger have the same definition for the connectivity space except that empty set is connected. Another terminology used for a c-structure is connectivity class [4,5,12] of X. We like to follow the terminology used in [7]. Elements of a c-structure are called connected sets. DX = {φ}∪{{x} : x ∈ X} is a c-structure on X, called the discrete c-structure and the space (X, DX ) is called the discrete c-space. Also, IX = P(X), the power set of X is a c-structure on X, called the indiscrete c-structure and the corre- sponding space is called an indiscrete c-space. A c-space (Y, CY ) is said to be a sub c-space of the c-space (X, CX ) if Y ⊆ X and CY = {A ∈ CX : A ⊆ Y }. Let B ⊆ P(X), the power set of X. Then the smallest c-structure on X containing B is called the c-structure generated by B and is denoted by < B >. The c-space (X, CX ) is denoted by X if there is no ambiguity. Let X and Y be two c-spaces and f : X → Y be a function. f is called c- continuous [7] or catenuous [7] or a connectivity morphism [4,12] or a connectivity map [3], if it maps connected sets of X to connected sets of Y . In [7], it is proved that the restriction of a c-continuous function and composition of two c-continuous functions are c-continuous. Also, a bijection f is said to be a c-isomorphism or catenomorphism if both f and f −1 are c-continuous. In [11], it is proved that if f : X → Y is a bijection, then f is a c-isomorphism if and only if f(CA)= CB. Let {Xi : i ∈ I} be a family of c-spaces and {fi : X → Xi : i ∈ I} be a family of functions defined on a set X. Let C be a collection of subsets of X such that A ∈ C C C if and only if fi(A) ∈ Xi for every i. Then is a c-structure on X and is called the strong c-structure generated by the given family of functions [11] and is denoted by < {fi : i ∈ I} >S . The c-structure on the product space Xi is the strong iQ∈I c-structure generated by the family of projection functions {πi : i ∈ I}. Hence the connected subsets of Rn are precisely those subsets of Rn whose projections are intervals. For example, the following figures represents some connected subsets and a disconnected subset of R2. 100 Santhosh P.K. A chain of connected sets [7] is a map C from a finite index set I to CX such that each Ci intersects its successor Ci+1, where Ci = C(i) for i ∈ I. Proposition 2.1. [1,7] Let B be a collection of subsets of a set X and let < B > be a c-structure on X generated by this collection(That is, the smallest c-structure on X containing B). Then the non trivial connected sets in < B > are characterized by the condition that any two points of such a connected set C can be joined by a finite chain of basic connected sets( i.e. elements of B) in C. That is, for all x, y ∈ C, we can find elements Bi, i = 0 to n in B such that Bi ⊆ C, Bi ∩ Bi+1 6= φ for i = 0 to n − 1 and x ∈ B0, y ∈ Bn. 3. Weak c-structure Generated by a Family of Functions Quotient spaces plays a vital role in all branches of mathematics. Though not stated in a formal language, the smallest c-structure on a set which makes a family of functions c-continuous can be found in [1] and for a single function in [3]. Here we are stating it formally as weak c-structure generated by a family of functions and quotient spaces are introduced as a particular case of this. Formalizing the concepts found in [1,3], we state the following definitions.