CutPoints and Separations in Alpha- Connected Topological Spaces

This paper introduces cutpoints and separations in -connected topological spaces, which are constructed by using the union of vertices set and edges set for a connected graph, and studies the relationships between them. Furthermore, it generalizes some new concepts.


1-Introduction
The concepts of -open sets and -closed sets in topological spaces were defined and studied firstly in [1] 1965 by O. Njastad. After this study, many mathematicians have generalized and derived other definitions in there researches. These studies were developed and many new relationships between these terms were found [2,3]. In general topology, the important concepts of cutpoints and cutpoint spaces have been studied in connected topological spaces [4][5][6][7][8][9]. Also, separations are associated with the term of cutpoints. Many references have taken these terms and discussed the relationships between them [10,11].
Throughout the last decades, the graph theory has been an essential part of combinatorial applications. The fundamental ideas appeared by Euler in 1736, when he solved a problem by introducing a graph that he constructed [12]. Thereafter, the graph theory became an important part of mathematics. As the applications have been increasingly appearing in multiple aspects in mathematics, there has been a growing interest in this concept [13,14]. Our research topic has further relations with additional previous works [15,16]. In this paper, we introduce the concept of cutpoints, with the topological viewpoint in -connected topological spaces, and study separations and connectedness in the -topological spaces. Finally, we prove some relationships and give a counter example.

Preliminaries and basic definitions
We need to recall some basic topological definitions and remarks with some definitions and facts for a graph. Let (X, ) be any topological space and A be a subset of X. If A satisfied the condition A Int(Cl(Int(A))) for all A X, then A is called an -open set. The set of all -open sets form -topology of X, which is denoted as . So, the pair (X, ) is called an -topological space, and the complement of A is an -closed set or Cl(Int(Cl(A))) A. The interior of these spaces is denoted as -(A), which means the union of all -open sets containing A, and the closure is denoted as -Cl(A), which means the intersection of all -closed sets contained in A. We conclude, by definition, that every open set is -open, but the converse is not true in general. We must refer to our topological space that it's structured under special conditions about the element of the set that built it. If G is any graph with vertices set and edges set, then we take the set X = and define on it, then the pair (X, ) is called topological space of the graph G with ground set , and satisfyied the conditions of general topology [11].  [11]. So, the separation on the subspace of the topological space X is defined as follows. Let Y be a subspace of a topological space X, then A, B are mutually separated in X if and only if there is a disjoint bipartition {A, B} which is a separation of Y related to the relative topology of Y [11]. Proposition 2.3 [11]: If A, B are two subsets of X and C is a connected subset of A B. If A, B are mutually separated subsets of a topological space X, then either C A or C B. Definition 2.4 [11]: If X is a topological space, then A, B, and C are disjoint subsets of X. If there exists a separation {H, K } of X \ A with B H and C K, then A separates B and C. We can use the same discussion when the subsets are singletons. Definition 2.5 [9]: If X is a topological space and A is a subset of X, then the intersection of Cl(A) and Cl(X -A) is called the boundary set of A. Definition 2.6 [9]: If X is any topological space, then the component of x is the largest connected subset of X containing x, and denoted as K(x). Proposition 2.7 [11]: Let X be a topological space and K be a component of X. If {U, V} is a separation of X, then K is contained either in U or in V. Definition 2.8 [4]: Let X be a connected topological space and x be any point in X. If X is not connected, then x is called a cutpoint of X. Otherwise, x is a non-cutpoint. Definition 2.9 [11]: Let X be a connected topological space and x be any point in X. If x is a cutpoint, then it is called a cutedge when it is an edge. However, it is called an endpoint if it is a non-cutpoint. Definition 2.10 [11]: Let X be a topological space and B X, then the intersection of all open sets that contain B is called the surrounding set of B, and denoted by . Definition 2.11 [11]: Let X be a topological space, then every point in X which is open but not closed is called hyperedge. When the boundary of the hyperedge contains exactly one point, it is called a loop. However, when its boundary contains at most two points, it is called an edge. Otherwise, it is a proper edge. We refer to the endvertex v, that means that there is only one edge that is incident on v [11]. Theorem 2.12 [11]: Let X be a connected topological space, x is a cutpoint of X, then:   [11]: Let X be a connected topological space and x be a cutpoint of X. If { , } is a separation of X \ {x}, then {x} is connected for i = 1, 2. Theorem 2.14 [11]: Let X be a connected topological space and h be a hyperedge of X. If { , , . . . } is a finite collection of non-empty closed subsets of X, when it is a partition of X \ {h}, then every part contains a point in the boundary of x.

3.Alpha-CutPoints And Alpha-Separations
Through this section, we introduce some concepts on -topological space and study their properties. Definition 3.1: An -topological space X is called separated space if there are unordered two disjoint subsets of X, such that each one is the complement of the other. The next proposition shows the relationship between being an -separating point and being andisconnect point. Proposition 3.6: Let X be an -topological space and x, y, z ∈ X. If y -separates x and z, then ydisconnects x and z. Proof: Since y -separates x from z, then there exists an -separation {U, V} of X \ {y} such that x ∈ U, z ∈ V. When the -component of X \ {y} is containing x were the same as the one containing z, then we obtain a non-empty intersection with both U and V, which is a contradiction with proposition (3.4). The following definition is a generalization of the cutpoint definition. Definition 3.7: If X is an -connected topological space and x is any point in X, then x is called ancutpoint of X if X is -disconnected. If not, then x is a non -cutpoint. The following theorem is a generalization of theorem (2.1.8) [11]. Proof: Since the -topology is a topology, the proof is similar to (2.1.8) [11].
The following theorem is a generalization of theorem (2.1.9) [11]. ∅ . The following definition is a generalization of the surrounding set. Definition 3.12: Let X be an -topological space and B X , the intersection of all -open sets containing B is called -surrounding set of B, denoted by . We can express the last two theorems in another way by using the above definition, as we see in the next proposition. Proposition 3.13: Let X be an -connected topological space and x is an -cutpoint of X, then one of the next statements is holding for any -separation { , } of X \ {x}: