From Topology to Anti-reflexive Topology

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1 From Topology to Anti-reflexive Topology Tsau Young ( T. Y. ) Lin, Guilong Liu, Mihir K. Chakraborty and Dominik Ślȩzak Department of Computer Science, San Jose State University San Jose, CA , USA School of Information Science, Beijing Language and Culture University Beijing , China Centre for Soft Computing Research, Indian Statistical Institute 204 B.T. Road, Kolkata, India Faculty of Mathematics, Informatics and Mechanics, University of Warsaw ul. Banacha 2, Warsaw, Poland Infobright Inc. ul. Krzywickiego 34/219, Warsaw, Poland tylin@sjsu.edu; liuguilong@blcu.edu.cn; mihirc4@gmail.com; slezak@mimuw.edu.pl Abstract A topological space is a space, where near makes sense; it is formally defined by the Topological N eighborhood System (TNS). Here, we explore the concept of conflict by the system of Anti-TNS ; by that we mean a mathematical structure that consists of a set of punctured neighborhoods, namely, the center point p of all neighborhoods of TNS has been removed. Conflicts are important concepts in computer security. The primary results is the axiomatization of ATNS. The main results are surprising: The set of the axioms is the same as that of topological spaces. Similar results for pretopological spaces also are obtained. Index Terms Neighborhood Systems, Granular Computing, Anti-reflexive Topology, Rough Sets I. Introduction The term neighborhood system (NS) in mathematics usually denotes topological neighborhood system (TNS). However, the NS that we are dealing with in this paper has been defined to its ultimate generality of TNS [1], [2]. For example, a neighborhood N(p) of p may be punctured or empty; by that we mean the neighborhood does not contain its center p or is an empty set. Such a neighborhood is called an anti-reflexive neighborhood, including the case of empty neighborhood. It is useful in many applications, e.g., in computer security. We may consider a set of my enemies as a neighborhood. Surely, myself is not included in that set [3], [4], [5]. Given big data applications, where big data is referred to as a set of data that is so large that any brute force combinatorial analysis is practically impossible, we shall focus on modeling NS on non-finite sets. Surely, nonfinite may have different interpretations for different applications. Sometimes it may mean that the universe is The work of the second author is supported by the National Natural Science Foundation of China (Nos and ). The work of the fourth author is supported by the Polish National Science Centre (Nos. 2011/01/B/ST6/03867 and 2012/05/B/ST6/03215). finite but its size makes it practically infinite for classical methods. In other cases, it may mean that we could not access the whole universe at one moment, e.g., in the applications related to data streams. In this paper, we follow one of recent developments in the area of NS axiomatization of NS in a similar sense as point free topology. An NS-space can be categorized by means of six types of derived sets, called pseudoderived sets [2]. One may say that such characterization follows an attempt of Sierpiński, described in 1952 [6]. In this particular paper, we apply this recent axiomatization of NS to various specific topologies. II. Category The Universe of Discourse Based on MacLane s idea [7], we shall use the category theory to formulate the universe of discourse in this paper. Roughly a category consists of two components [8]: 1) A class of objects. 2) A class of morphisms: A set Mor(X, Y ) of morphisms, for every ordered pair of objects X and Y, that satisfies certain conditions. Here are some simple examples: 1) Category of sets: The class of objects is the class of (crisp) sets. Mor(X, Y ) is the set of mappings from X to Y. 2) Category of topological spaces: The class of objects is the class of topological spaces. Mor(X, Y ) is the set of continuous mappings from X to Y. III. Topology (TNS) and Rough Sets (RS) Let us recall the basics of topology. Next paragraphs are taken from [9]. A topology is a collection τ of sets satisfying the two the intersection of finitely many members of τ is a member of τ, and the union of the members of any subcollection of τ is a member of τ. Any member of τ is called an open set. The union

2 U = {O : O τ} is necessarily a member of τ because τ is a subfamily of itself, and every member of τ is a subset of U. The pair (U, τ) or simply U (when context is clear) is called the topological space of the topology τ, and τ is a topology of the topological space U. A set N in a topological space (U, τ) is called a τ- neighborhood (or simply neighborhood) of p U iff N contains an open set to which p belongs. The collection T NS(p) of all such N is called the topological neighborhood system (TNS) at p. One can use the family T N S(U) = {T N S(p) : p U}, called a topological neighborhood system of U, to define the topological space (U, τ). We shall take this approach: Definition 1: (Topological Space) The pair (U, T N S(U)) is called a topological space (or TNSspace), if T NS(U) = {T NS(p) : p U}, where, for each p U, T NS(p) is the family of all subsets, called neighborhoods, that satisfy the following axioms: ([9], Ch 1, Exercise B) 1) If N T NS(p), then p N. 2) If N and M are members of T NS(p), then N M T NS(p). 3) If N T NS(p) and N M, then M T NS(p). 4) If N T NS(p), then there is M T NS(p) such that M N and M T NS(y) for each y in M (that is, M is a neighborhood of each of its points). Definition 2: (Base) A base for T NS(p) of a point p is a family of neighborhoods such that every neighborhood N T NS(p) contains a member of the family. Definition 3: (TNS-continuous Mappings) Let T NS 1 (U 1 ) = {T NS 1 (p) p U 1 } and T NS 2 (U 2 ) = {T NS 2 (q) q U 2 } be two neighborhood systems on U 1 and U 2, respectively. The mapping f : U 1 U 2 is said to be: 1) Continuous at a point p U, if q = f(p), nonempty N T NS 1 (p), satisfies f(n) M, nonempty M T NS 2 (q). 2) Continuous iff f is continuous at every p U. The topological spaces and their continuous mappings form the category of TNS-spaces (topological spaces). Example 1: Rough set theory (RS) is very closely related to the concept of topology from the very beginning. As pointed out in [10], the approximation space is a topological space. The upper and lower approximations are then actually closure and interior. For infinite universes, one can further extend the admissible knowledge to finite and and arbitrary or (observe that one does not need to follow Pawlak to use composed sets as topology). One can also investigate connections between RS and TNS using a variety of algebraic and geometrical models [11], [12], which can be all interpreted using the fundamental framework of neighborhood systems. Let R be an equivalence relation on U. The rough set approximation space is a clopen topological space whose topology is generated by the equivalence classes of R. Consider U = {1, 2, 3, 4}. Assume it is partitioned into subsets {1, 2} and {3, 4}, which generate the topology. The topology (the collection of all open sets) is: A base of TNS is: {, {1, 2}, {3, 4}, {1, 2, 3, 4}} NS(1) = {{1, 2}}, NS(2) = {{1, 2}}, NS(3) = {{3, 4}}, NS(4) = {{3, 4}}. Observe that the third axiom basically says all supersets of a neighborhood should be included, so N S(p) gets expanded: T NS(1) = {{1, 2}, {1, 3, 2}, {1, 2, 4}, {1, 3, 2, 4}} T NS(2) = {{1, 2}, {1, 3, 2}, {1, 2, 4}, {1, 3, 2, 4}} T NS(3) = {{3, 4}, {1, 3, 4}, {2, 3, 4}, {1, 3, 2, 4}} T NS(4) = {{3, 4}, {1, 3, 4}, {2, 3, 4}, {1, 3, 2, 4}} It is easy to verify that this set meets the conditions in Definition 1. In summary, in this example we showed that the approximation space can be defined by: (1) the collection of open sets, (2) a base of TNS, and (3) TNS. IV. Category of Neighborhood Systems (NS) The category of neighborhood systems (NS) will be treated as the universe of discourse. Let U be a (not necessarily finite) classical set, P (U) be the family of all crisp or fuzzy subsets, and 2 Y be the family of all crisp subsets of Y, where Y = P (U). In the rest of the paper we will skip the adjective crisp or fuzzy, if the context is clear. However, the presented results can be considered in both cases. Definition 4: (LNS-spaces) Let NS(p) = {N(p) U} be a family of subsets of U that is associated with each p U. The pair (U, NS(U)) is called an LNS-space. It is called the Local/First GrC Model in [13]. A member (could be empty) of NS(p) is a neighborhood of p; p is the center of (every member of) NS(p). Example 2: BNS-space is a special NS-space defined by a binary relation BR on U. The object is defined by: BNS(U) = {BNS(p) p U} BNS(p) = {y (p, y) BR} The pair (U, BNS) is called a BNS-space; it is called Binary/Third GrC Model in [13]. Let U = {a, b, c}, and B be a binary relation defined by the alphabetical order: a b c. Then its BNS is: a {{a, b, c}} b {{b, c}} c {{c}}

3 It is easy to see that BNS-space is a topological space; The open sets are: {a, b, c}, {b, c}, {c} Its TNS is expanded into: a {{a, b, c}} b {{b, c}, {a, b, c}} c {{c}, {a, c}, {b, c}, {a, b, c}} Observe that, in general, a BNS is not a TNS. Definition 5: (NS-continuous Mappings) Let NS 1 (U 1 ) = {NS 1 (p) p U} and NS 2 (U 2 ) = {NS 2 (q) q U 2 } be two neighborhood systems on U 1 and U 2, respectively. The mapping f : U 1 U 2 is said to be: 1) Continuous at a point p U 1, if any of the following is true: a) If N S(q) and N S(p), for q = f(p), have non-empty neighborhoods, then for every nonempty M N S(q) there is a non-empty N NS(p) such that N is mapped by f into M (i.e., f(n) M). b) N S(p) has empty neighborhood only. c) NS(q) is an empty family. 2) Continuous iff f is continuous at every p U 1. The NS-spaces and their continuous mappings form the Category of Neighborhood Systems, which is the universe of discourse in this paper. Definition 6: (NS-equivalence) 1) The mapping f : U 1 U 2 is said to be topologically equivalent or NS-equivalent iff f 1 exists and both f and f 1 are continuous. 2) Two NS-spaces (U, NS 1 (U)) and (U, NS 2 (U)) are topologically equivalent or NS-equivalent iff for every neighborhood N 1 of a point p in (U, NS 1 (U)) there exists a neighborhood N 2 of that point p in (U, NS 2 (U)) which is contained in N 1, and vice versa. They are also said to process the same topological structure or, more briefly, the same topology. From the categories viewpoint, (U, NS 1 (U)) and (U, NS 2 (U)) are the same object. Definition 7: (Canonical Model of NS) 1) An NS is called the largest neighborhood system (LNS), if it satisfies the following superset condition at each p: If M N(p) for some non-empty N(p) NS(p), then M NS(p). 2) In the case of topological spaces, such an LNS is the topological neighborhood system (TNS), which is defined axiomatically in Section II. Proposition 1: LNS(U) is the largest NS(U) among all the topologically equivalent NS s on U. Proof: See Proposition 3.2 in [14], QED. V. Simple Neighborhood Systems of Type K For axiomatization purposes, we have decomposed an NS to a union of simple neighborhood systems (Simple- NS) [2]. Definition 8: (Simple-NS) Let S U. An NS(U) is called a Simple-NS of type K on S or simply K-NS(S) if 1) NS(p) is a K-NS on all p S; such p is called K-NS point. 2) NS(p) is equal to the empty family if p / S. There are six types of Simple-N S(S): Definition 9: (K-NS s) 1) N S(S) is the empty initial neighborhood system (EI-NS) on U if NS(p) = { }, p S, and empty families elsewhere. 2) N S(S) is the empty terminal neighborhood system (ET-NS) on U if NS(p) is an empty family, p S, and empty families elsewhere. (Observe that we state this system in this way, because we want to adopt to the same representation of other systems. In fact, every NS(p) is an empty family.) 3) N S(S) is the pure anti-reflexive neighborhood system (PA-NS) on U if NS(U) consists of a nonempty family N S(p) of non-empty anti-reflexive neighborhoods on S and empty families elsewhere. 4) N S(S) is the pure reflexive neighborhood system (PR-NS) on U if NS(U) consists of a non-empty family NS(p) of reflexive neighborhoods on S and empty families elsewhere. 5) NS(S) is the anti-reflexive empty neighborhood system (AE-NS) on U if NS(U) consists of an empty neighborhood ( is a neighborhood) and a nonempty family of non-empty anti-reflexive (punctured) neighborhoods on S and empty families elsewhere. 6) N S(S) is the reflexive empty neighborhood system (RE-NS) on U if N S(U) consists of an empty neighborhood and a non-empty family of reflexive neighborhoods on S and empty families elsewhere. Observe that in general the domain S of these six simple NS of type K is on the same U with distinct S. VI. Pseudo-derived Sets of Simple-NS Next, we will port the concept of limit points, derived sets, closure and interior from TNS to NS. Roughly, if an NS-system has empty neighborhoods, then a derived set is called a pseudo-derived set. Definition 10: (Pseudo-Derived Sets and Derived Sets) Let NS(U) = {NS(p) p U}, where for every p U, NS(p) = {N(p) U} is a given Simple-NS on U, and X U. 1) There are no pseudo-derived sets for any subsets of ET-NS points. 2) The pseudo-derived set of X = is the set of EI-NS points.

4 3) A point p U is called a pseudo-limit point of nonempty X if p has non-empty neighborhoods and each of its non-empty neighborhoods contains at least one point of X other than p. That is, p is a pseudo-limit point of X, if NS(p) contains at least one point of X other than p and for every non-empty N NS(p), N (X {p}). The pseudo-derived set of X, denoted by X, is the set of all pseudo-limit points of X. 4) A point p U is called a limit point of non-empty X, if it is a pseudo-limit point of X and / NS(p). The derived set of X, denoted by X, is the set of all limit points of X. 5) The derived set of X = is the empty set. That is, =. The pseudo-derived set, D p (X), of Simple-NS of type K is denoted D K (X). Each NS-space has six types of derived sets, that is K = EI, ET, PR, PA, AR, AE. Observe that Definition 10 has many hidden points. 1) The pseudo-derived set and derived set of the empty set are defined directly. 2) If NS(U) is an EI-NS, then X = for any nonempty X U and = {EI-NS points}. 3) If NS(U) is an RE-NS, then the derived set X (but not pseudo-derived set X ) is empty set for any X U. 4) If NS(U) is a PR-NS, then derived set and pseudoderived set are equal, that is, X = X for any non-empty X U. Definition 11: (NS-closure) The concept of NS-closure is similar to pseudo-limit points: we only drop the requirement other than p itself. That is: 1) There is no NS-closure for any subsets of ETpoints. 2) The NS-closure of X = is the set of EI -points. 3) A point p U is in NS-closure of non-empty X if p has non-empty neighborhoods and each of its nonempty neighborhoods contains at least one point of X. 4) If p has an empty neighborhood only, then p is in NS-closure of X =. Such NS-closure is denoted by C(X). Observe that such NS-closure, denoted by C(X), may not be a closed set; In earlier works [14], TNSterminology was used for NS-terminology; some of them may be confusing. C(X) should be termed pre-closure and should be said to be pre-closed set. The real closed set was called closed set-based closure later. Definition 12: (NS-Interior) Let U be a given NSspace. An element p of U is said to be in the interior of the set X if there is a non-empty N NS(p) such that p N and N X. The set of all such elements of a set X is called NS-interior of X and is denoted by I(X). A. Simple Examples Example 3: Consider an ATNS by puncturing every neighborhood of TNS (topology) in Example 2. For U = {a, b, c}, NS is given by the following equations: Its LNS is: a {{b, c}} b {{c}, {a, c}} c {{}, {a}, {b}, {a, b}} a {{b, c}, {a, b, c}} b {{c}, {a, c}, {b, c}, {a, b, c}} c {, {a}, {b}, {a, b}, {a, c}, {b, c}, {a, b, c}} Observe that its reflexive closure is the original TNS, namely, Example 2. ATNS is a PA-NS, so the only non-empty derived sets are derived sets of PA-NS, denoted by: D P A (a) = D P A (b) = {a} D P A (c) = {a, b} D P A (a, b) = {a} D P A (a, c) = {a, b} D P A (b, c) = {a, b} and all other types of derived sets are empty. Let us compute the derived sets of Example 2, which is a TNS. It is a PR-NS, so the only non-empty derived sets are: Example 4: D P R (a) = D P R (b) = {a} D P R (c) = {a, b} D P R (a, b) = {a} D P R (a, c) = {a, b} D P R (b, c) = {a, b} and all other types of derived sets are empty. The ATNS and TNS have the same sets of derived sets but in different types.

5 VII. Special NS-spaces Definition 13: (Reflexive Closure of NS) Let B be an NS on U. 1) B is said to be reflexive, if p U, if N i (p) NS(p) then p N i (p). 2) B is the reflexive closure of B, if: a) For every neighborhood N i (p) B(p), there is a unique reflexive neighborhood N i (p) B (p), such that N i (p) N i(p). That can be abbreviated as B B. b) For any reflexive NS B, if B B then B B. The reflexive closure is easy to construct. One needs to add the center to each neighborhood, namely, N new (p) N old (p) {p}. Definition 14: (Anti-reflexive Closure of NS) Let B be an NS on U. 1) B is said to be anti-reflexive, if p U, if N i (p) NS(p) then p / N i (p). 2) B is the anti-reflexive closure of B, if: a) For every neighborhood N i (p) B(p), there is a unique anti-reflexive neighborhood N i (p) B (p) such that N i (p) N i(p). That can be abbreviated as B B. b) For any anti-reflexive NS B, if B B then B B. The anti-reflexive closure is also easy to construct. One needs to remove the center for each neighborhood, namely, N new (p) N old (p) {p}. Definition 15: (Transitive Closure of NS) Let B be an NS on U. 1) B is said to be transitive, if p, q, r U, if p N i (q) and q N i (r) then p N i (r), where N i (x) B(x), for x = p, q, r. Using geometrical language, B is an open NS, that is every neighborhood N i contains a neighborhood of every x N i. 2) B is the transitive closure of B, if: a) For every neighborhood N i (p) B(p), there is a unique transitive (open) neighborhood N i (p) B (p), such that N i (p) N i(p). That can be abbreviated as B B. b) For any transitive NS B, if B B then B B. Transitive closure can be obtained by transfinite applications of NS-closure C( ) to every neighborhood. Definition 16: (Anti-transitive Closure of NS) Let B be an NS on U. Let C B be an NS that consists of the compliment of every N i (p) B(p), namely, M i (p) is the compliment of N i (p). The collection of such M i (p) forms C B (p). 1) B is said to be anti-transitive, if the compliment C is transitive, namely, if p, q, r U, if p M i (q) and q M i (r) then p M i (r), where M i (x) C B (x), for x = p, q, r. 2) B is the anti-transitive closure of B, if: a) For every neighborhood N i (p) B(p), there is a unique anti-transitive neighborhood N i (p) B (p) such that N i (p) N i(p). That can be abbreviated as B B. b) For any anti-transitive NS B, if B B then B B. Anti-transitive closure can be obtained as follows: First, one needs to find the transfinite closure of C B ( ) as in Definition 15. Then one can define B new (p) to be its compliment. A. Anti-reflexive Topology Let us consider an NS-space, in which the NS is obtained by applying the anti-reflexive closure to a TNS, namely, it consists of punctured topological neighborhoods, that is, the center p is removed from every neighborhood N(p) T NS(p); recall that TNS is abbreviation of topological neighborhood system. Definition 17: (Anti-reflexive Topological Space) The NS-space (U, AT N S(U)) is called an anti-reflexive topological space (or ATNS-space), if AT N S(U) = {AT NS(p) : p U}, where, for each p U, AT NS(p) is the family of all subsets of U satisfying the following axioms: 1) If N AT NS(p), then p / N. 2) If N and M are members of AT NS(P ), then N M AT NS(p). 3) If N AT NS(p) and N M {p}, then M {p} AT NS(p). 4) If N AT NS(p), then there is a member M of AT NS(p) such that M N and (M {p}) {y} AT NS(y) for each y in (M {p}) (that is, M {p} {y} is a neighborhood of each of its points y). ATNS is a PA-NS. Proposition 2: Reflexive closure of an ATNS-space is a TNS-space. Conversely, anti-reflexive closure of a TNS-space is an ATNS-space. B. Anti-/Pretopology and Closure Spaces Pretopology is an ancient concept introduced by Hausdorff [15]. Definition 18: (Pretopological Space) The pair (U, P NS(U)) is called a pretopological space (or PNSspace), if P NS(U) = {P NS(p) : p U}, where, for each p U, P NS(p) is the family of all subsets, called neighborhoods, satisfying the following axioms: 1) If N P NS(p), then p N. 2) If N and M are members of P NS(P ), then N M P NS(p). 3) If N P NS(p) and N M, then M P NS(p). Pretopology is a PR-NS. By applying anti-reflexive closure to PNS, we have APNS.

6 Definition 19: (APNS-space) The pair (U, AP N S(U)) is called APNS-space, if AP NS(U) = {AP NS(p) : p U}, where, p U, P NS(p) is the family of all subsets, called neighborhoods, satisfying the following axioms: 1) If N AP NS(p), then p / N. 2) If N and M are members of P NS(P ), then N M AP NS(p). 3) If N AP NS(p), N M and p / M, then M AP NS(p). APNS-space is a PA-NS. Proposition 3: In a PNS-space or APNS-space, (X Y ) X Y. Proof: Let p D(X Y ), and assume p / D(X) D(Y ), that is, an N(p) (X {p}) =, or an N (p) (Y {p}) =. That is, (N(p) N (p)) ((X {p}) (Y {p})) =. Observe that N(p) N (p) P NS(p), that is, p / D(X Y ), QED. Definition 20: (Closure Space) A closure space is a pair (U, C), where closure operator C : P (U) P (U) is a function that associates each subset X U with a subset C(X) U, called the closure of X, such that 1) C( ) =. 2) C(X Y ) = C(X) C(Y ). Definition 21: (Anti-closure Space) Anti-reflexive closure of a closure space is called an anti-closure space. Proposition 4: A PNS-space and APNS-space are a closure and anti-closure spaces, respectively, and vice versa. Proof: Observe that in PNS-space C(X) = X D p (X), and in APNS-space C(X) = D p (X) so 1) C( ) =. 2) C(X Y ) = C(X) C(Y ). We can put NS(p) = {H p / C(U \ H)}. This family defines a PNS-space [15]. Similar argument works for anti-spaces, QED. Definition 22: (Transitive Closure and Anti-closure Spaces) Transitive closure of a closure space (and anticlosure) is called transitive closure space (and transitive anti-closure space), denoted as TC-space (and TACspace). Proposition 5: Both TC-space and TAC-space of a BNS-space are BNS-spaces. Proof: We can adopt the proof of Theorem 4.1 in [16] to TAC-spaces, QED. VIII. Axiomatizations of Special NS A. Axiomatizations of PA-NS and PR-NS From [2], we note that for PR-NS, we have: Theorem 1: Let D p : P(U) P(U) assign to each 1) D p ( ) = {EI points}. Then, one can define neighborhoods of the elements of U so that U is a PR-NS, where X = D p (X), X U. For pure anti-reflexive neighborhood systems, we have: Theorem 2: Let D p : P(U) P(U) assign to each 1) D p ( ) = D T (U D p (U)). Then, one can define neighborhoods of the elements of U so that U is a PA-NS, where X = D p (X), X U. B. Axiomatizations of PNS and APNS Theorem 3: Let D p : P(U) P(U) assign to each 4) D p (X Y ) D p (X) D P (Y ). Then, one can define neighborhoods of the elements of U so that U is a PNS, where X = D p (X), X U. Theorem 4: Let D p : P(U) P(U) assign to each 4) D p (X Y ) D p (X) D P (Y ). Then, one can define neighborhoods of the elements of U so that U is an APNS, where X = D p (X), X U. C. Axiomatizations of TNS and ATNS Theorem 5: Let D p : P(U) P(U) assign to each 2) D p (D p (X)) D p (X). 3) D p (X) is monotonic on non-empty sets. 4) D p (X Y ) D p (X) D p (Y ). 5) If x D p (X) then x D p (X {x}). Then, one can define neighborhoods of the elements of U so that U is a TNS, where X = D p (X), X U. The above facts can be proved by following a standard text book. For ATNS, we have: Theorem 6: Let D p : P(U) P(U) assign to each 2) D p (D p (X)) D p (X). 3) D p (X) is monotonic on non-empty sets. 4) D p (X Y ) D p (X) D p (Y ). 5) If x D p (X) then x D p (X {x}).

7 Then, one can define neighborhoods of the elements of U so that U is an ATNS, where X = D p (X), X U. This theorem follows immediately from the observation that the derived sets of TNS and ATNS are the same, given the definitions of limit points. IX. Conclusions Anti-reflexivity is very important in a number of areas of computer applications. In this paper, we axiomatized pretopological spaces that are reflexive and anti-reflexive. This may allow us to port many topological techniques to, e.g., the field of computer security. Moreover, since anti-reflexive topological neighborhood systems allow us to describe the infinitesimals (not including standard number zero) in standard and non-standard world, some non-reflexive phenomena of real number systems may be studied. References [1] T. Y. Lin, Neighborhood Systems and Approximation in Database and Knowledge Base Systems, in Proc. of ISMIS 1989, [2] T. Y. Lin and Y. R. Syau, Granular Mathematics Foundation and Current State, in Proc. of IEEE GrC 2011, [3] T. Y. Lin, Chinese Wall Security Policy An Aggressive Model, in Proc. of ACSAC 1989, [4], Chinese Wall Security Policy Models: Information Flows and Confining Trojan Horses, in Proc. of DBSec 2003, 2003, pp [5] T. Y. Lin and J. Pan, Granular Computing and Flow Analysis on Discretionary Access Control: Solving the Propagation Problem, in Proc. of IEEE SMC 2009, [6] W. Sierpiński, General Topology. University of Toronto Press, [7] S. MacLane, Homology. Academic Press, [8] E. H. Spanier, Algebraic Topology. McGraw-Hill, [9] J. Kelley, General Topology. Springer, [10] Z. Pawlak, Rough Sets, International Journal of Information and Computer Sciences, vol. 11, no. 5, pp , [11] P. Pagliani and M. Chakraborty, A Geometry of Approximation, ser. Trends in Logic. Springer, 2010, vol. 27. [12] P. Wasilewski and D. Ślȩzak, Foundations of Rough Sets from Vagueness Perspective, in Rough Computing Theories, Technologies and Applications, A. E. Hassanien, Z. Suraj, D. Ślȩzak, and P. Lingras, Eds. IGI Global, 2008, pp [13] T. Y. Lin, Granular Computing: Practices, Theories, and Future Directions, in Encyclopedia of Complexity and Systems Science, 2009, pp [14], Topological and Fuzzy Rough Sets, in Decision Support by Experience Application of the Rough Sets Theory, R. S lowiński, Ed., 1992, pp [15] F. Hausdorff, Gestufte Räume, Fundamenta Mathematicae, vol. 25, no. 1, pp , [16] G. Liu, Closures and Topological Closures in Quasi-discrete Closure Spaces, Appl. Math. Lett., vol. 23, no. 7, pp , 2010.