Connectivity with respect to α-discrete closure operators

• Josef Šlapal
From the journal Open Mathematics

Abstract

We discuss certain closure operators that generalize the Alexandroff topologies. Such a closure operator is defined for every ordinal α > 0 in such a way that the closure of a set A is given by closures of certain α -indexed sequences formed by points of A . It is shown that connectivity with respect to such a closure operator can be viewed as a special type of path connectivity. This makes it possible to apply the operators in solving problems based on employing a convenient connectivity such as problems of digital image processing. One such application is presented providing a digital analogue of the Jordan curve theorem.

MSC 2010: 54A05; 54D05

1 Introduction

One of the basic problems of digital topology is to equip the digital plane Z 2 with a convenient connectivity, i.e., a connectivity behaving analogously to the connectivity in the Euclidean plane R 2 . In particular, such a connectivity is required to satisfy a digital analogue of the Jordan curve theorem because digital Jordan curves represent borders of objects in digital pictures. In this note, we study certain closure operators (more general than the Kuratowski ones) and show that they provide a convenient connectivity for the digital plane.

By a closure operator u on a set X , we mean a map u : exp X exp X (where exp X denotes the power set of X ), which is

1. grounded (i.e., u = ),

2. extensive (i.e., A X A u A ), and

3. monotonic (i.e., A B X u A u B ).

The pair ( X , u ) is then called a closure space.

These closure operators were studied in the pioneering paper [1] by Čech published as early as in 1936.

Recall that a closure operator u on X which is

1. additive (i.e., u ( A B ) = u A u B whenever A , B X ) and

2. idempotent (i.e., u u A = u A whenever A X )

is called a Kuratowski closure operator or a topology, and the pair ( X , u ) is called a topological space.

In the literature, closure operators are usually understood to be idempotent and not necessarily grounded, while the ones introduced above are named preclosure operators. But, in this note, we follow the terminology of categorical topology where closure operators in the above sense are commonly used.

Given a cardinal m > 1 , a closure operator u on a set X and the closure space ( X , u ) are called an S m -closure operator and an S m -closure space (briefly, an S m -space), respectively, if the following condition is satisfied:

A X u A = { u B ; B A , card B < m } .

The closure operators usually employed in algebra are the idempotent ones. The so-called algebraic closure operators are then simply the idempotent S 0 -closure operators (cf. [2]).

In the categorical topology, an important role is played by the closure operators on categories studied, e.g., in [3]. They generalize the above closure operators by considering them on subobject lattices in a given category (instead of on power sets, i.e., the subobject lattices in the category S e t ).

In [4], (additive) S 2 -closure operators and S 2 -spaces are called quasi-discrete. S 2 -topologies ( S 2 -topological spaces) are usually called Alexandroff topologies (Alexandroff spaces) [5]. Alexandroff topologies play an important role in general topology and many other branches of mathematics, particularly in general and topological algebra (Scott topologies, monoid actions, commutative algebra) and computer science (Khalimsky topology), cf. [6,7,8, 9,10,11]. Of course, every S 2 -closure operator is additive, and every S α -closure operator is an S β -closure operator whenever α < β . Since any closure operator on a set X is obviously an S α -closure operator for each cardinal α with α > card X , there exists a least cardinal α such that u is an S α -closure operator. Such a cardinal is then an important invariant of the closure operator u . Evidently, if α 0 , then every additive S α -closure operator is an S 2 -closure operator.

We will use some known topological concepts (see, e.g., [12]) naturally extended to closure spaces. Given a closure space ( X , u ) , a subset A X is called closed if u A = A , and open if X A is closed. A closure space ( X , u ) is said to be a subspace of a closure space ( Y , v ) if u A = v A X for each subset A X . We will speak briefly about a subspace X of ( Y , v ) . A closure space ( X , u ) is said to be connected if and X are the only subsets of X to be both closed and open. A subset X Y is connected in a closure space ( Y , v ) if the subspace X of ( Y , v ) is connected. A maximal connected subset of a closure space is called a component of this space. All the basic properties of connected subsets and components in topological spaces are also preserved in closure spaces. In particular, if { A i ; i I } is a family of connected subsets with a non-empty intersection, then i I A i is connected, too.

In the paper, we will work with α -indexed sequences of points of a set X , where α is an ordinal, i.e., with sequences of the form ( x i i < α ) X α . We will write ( x i i α ) instead of ( x i i < α + 1 ) .

2 α -discrete closure operators

In the sequel, α > 0 will be an ordinal.

Definition 2.1

Let ( X , u ) be a closure space. A sequence ( x i i α ) X α + 1 is called a u-connected element if x j u { x i ; i < j } for each j , 0 < j α . The elements x 0 and x α are called the end points of the u -connected element ( x i i α ) .

Definition 2.2

A closure operator u on a set X is called α -discrete if the following condition is fulfilled:

For any A X and any x u A , there exists a u -connected element ( x i i α ) such that ( x i i < α ) A α and x α = x .

Thus, every α -discrete closure operator is an S α -space, where α denotes the least of all cardinals that are greater than α . The 1-discrete closure operators coincide with the S 2 -closure operators so that idempotent 1-discrete closure operators are simply Alexandroff topologies, i.e., topologies with completely additive closures.

Example 2.3

Let u be the closure operator on the set X = { a , b , c } given by u { a } = { a , b } , u { b } = { b } , u { c } = { c } , u { a , b } = u { a , c } = X , and u { b , c } = { b , c } . Then, u is a 2-discrete closure operator on X (which is not 1-discrete). This results from the fact that the sequences ( a , a , a ) , ( b , b , b ) , ( c , c , c ) , ( a , a , b ) , and ( a , b , c ) are u -connected elements and, for every A X and x u A , there exists ( x i i 2 ) { ( a , a , a ) , ( b , b , b ) , ( c , c , c ) , ( a , a , b ) , ( a , b , c ) } such that ( x i i < 2 ) A 2 and x 2 = x .

For a closure operator u on a set X , we denote by u α the closure operator on X given by u α A = { x X ; there is a u -connected element ( x i i α ) such that x = x i 0 for some i 0 , 0 < i 0 α , and x i A for all i < i 0 } .

Proposition 2.4

A closure operator u on a set X is α -discrete if and only if u = u α .

Proof

Let u be a closure operator on X . Let u be α -discrete and let A X be a subset. It is evident that u A u α A . To show the converse inclusion, let x u α A . Then, there is a u -connected element ( x i i α ) such that x = x i 0 for some i 0 , 0 < i 0 α , and x i A for all i < i 0 . Thus, x u { x i ; i < i 0 } u A . Therefore, u = u α .

Conversely, let u = u α . Let A X and x u A . Then, there exists a u -connected element ( x i i α ) such that x = x i 0 for some i 0 , 0 < i 0 α , and x i A for all i < i 0 . Put x i = x i for all i < i 0 , x i = x 0 for all i , i 0 i < α , and x α = x i 0 . Then, ( x i i α ) is a u -connected element such that ( x i i < α ) A α and x α = x . Therefore, u is α -discrete.□

Example 2.5

As usual, we denote by ω the least infinite ordinal. Let u be the closure operator on the set (ordinal) ω + 1 given by u A = { α ; α is an ordinal such that min A α ω } . Let x u A be a point. Then, the points y A with y < x form an increasing sequence ( y i i < α ) , where α ω , y 0 = min A , and x j u { x i ; i < j } for each j , 0 < j < α . Put y i = y i for all i < α and y i = x for all i , α i ω . Then, ( y i i ω ) is a u -connected element such that x = y α and y i A for all i < α . Therefore, u = u ω . By Proposition 2.4, u is an ω -discrete closure operator (which is not α -discrete for any finite ordinal α ).

Proposition 2.6

Let u be an α -discrete closure operator on a set X. Then:

1. The union of a system of closed subsets of ( X , u ) is a closed subset of ( X , u ) .

2. u is idempotent if and only if ( X , u ) is an Alexandroff space.

Proof

(1) Let { A j ; j J } be a system of closed subsets of ( X , u ) and let x u j J A j . Then, there exists a u -connected element ( x i i α ) such that x = x α and x i j J A j for all i < α . In particular, we have x 0 j J A j , and so there exists j 0 J such that x 0 A j 0 . Suppose that { x i ; i < α } is not a subset of A j 0 . Then, there is the smallest ordinal i 1 < α such that x i 1 A j 0 . Consequently, 0 < i 1 and x i A j 0 for all i < i 1 . Thus, we have x i 1 u { x i ; i < i 1 } u A j 0 = A j 0 , which is a contradiction. Therefore, { x i ; i < α } A j 0 , and hence, x u { x i ; i < α } u A j 0 j J u A j . We have shown that u j J A j j J u A j . As the converse inclusion is obvious, the proof is complete.

(2) Let u be idempotent, let A X and x u A . Then, there is a u -connected element ( x i i α ) such that ( x i i < α ) A α and x α = x . Thus, we have x j u { x i ; i < j } for all j , 0 < j α , and clearly, x 0 u { x 0 } . Let j be an ordinal, 0 < j α , such that x i u { x 0 } for all i < j . Then, x j u { x i ; i < j } u u { x 0 } = u { x 0 } . Therefore, by transfinite induction, x j u { x 0 } for every j α . Consequently, x = x α u { x 0 } . Hence, u is an Alexandroff topology. The converse implication is obvious.□

Definition 2.7

Let u be an α -discrete closure operator on a set X and x , y X . A finite sequence of u -connected elements ( p j ) j = 1 k = ( ( x i j i α ) ) j = 1 k ( k a positive integer) is said to be an α -path connecting x and y if

• x is an end point of p 1 with the other end point of p 1 coinciding with an end point of p 2 ,

• an end point of p j coincides with an end point of p j 1 and the other end point of p j coincides with an end point of p j + 1 for j = 2 , 3 , , k 1 , and

• y is an end point of p k and the other end point of p k coincides with an end point of p k 1 .

Thus, every u -connected element ( x i i < α ) is an α -path connecting x 0 and x α (and also an α -path connecting x α and x 0 ). Clearly, if ( p j ) j = 1 k is an α -path connecting x and y , then ( p k j + 1 ) j = 1 k is an α -path connecting y and x . It is also evident that any α -path (connecting a pair of points) ( ( x i j i α ) ) j = 1 k is a connected set, namely, the set j = 1 k { x i j ; i α } . If ( p j ) j = 1 k is an α -path connecting x and y and ( q j ) j = 1 l is an α -path connecting y and z , then the α -path ( r j ) j = 1 k + l , where r j = p j whenever 1 j k and r j = q j k whenever k < j k + l is an α -path connecting x and z .

Theorem 2.8

Let u be an α -discrete closure operator on a set X and A X a subset. Then, A is connected in ( X , u ) if and only if any two points of A can be joined by an α -path contained in A.

Proof

If A = , then the statement is trivial. Let A . In ( X , u ) , if any two points of A can be connected by an α -path, then A is clearly connected (because, choosing a point x A , we have y A { P y , P y is an α -path contained in A connecting x and y } = A , i.e., A is the union of connected sets with a non-empty intersection). Conversely, let A be connected and suppose that there are points x , y A that cannot be connected by an α -path contained in A . Let B be the set of all points of A that can be connected with x by an α -path contained in A . Let z u B A be a point. Then, there is a u -connected element ( x i i α ) such that ( x i i < α ) B α and x α = z . Hence, ( x i i α ) is a u -connected element contained in A , thus an α -path connecting the points x 0 B and z A . Since x and x 0 can also be connected by an α -path contained in A , so can x and z . Therefore, z B , i.e., u B A = B . Consequently, B is closed in the subspace A of ( X , u ) . Next, let z u ( A B ) A be a point. Then, there is a u -connected element ( x i i α ) such that ( x i i < α ) ( A B ) α and x α = z . Suppose that z B . Then, x can be connected with z by an α -path contained in A . Further, z can be connected with x 0 by an α -path – the u -connected element ( x i i α ) – contained in A . Consequently, x and x 0 can be connected by an α -path contained in A , which is a contradiction with x 0 B . Thus, z B , i.e., u ( A B ) A = A B implying that A B is closed in the subspace A of ( X , u ) . Hence, A is the union of the non-empty disjoint sets B and A B closed in the subspace A of ( x , u ) . But this is a contradiction because A is connected. Therefore, any two points of A can be connected by an α -path contained in A .□

It is well known [13] that closure operators that are more general than the Kuratowski ones have useful applications in computer science. By Theorem 2.8, connectivity with respect to an α -discrete closure operator is a certain type of path connectivity. This enables us to apply these closure operators in digital image processing because they provide connectivity structures suitable for studying the geometric and topological properties of digital images (cf. [14]). There are two well-known 1-discrete topologies on Z 2 employed in digital image processing, Marcus topology [15] and Khalimsky topology [16]. In [17], a 2-discrete closure operator on Z 2 is used to formulate and prove a digital analogue of the Jordan curve theorem – see the following example.

Example 2.9

Let u be a closure operator on Z 2 and G be a simple undirected graph (without loops) with the vertex set Z 2 . A circuit in G is said to be a simple closed curve in G with respect to u if it is a minimal (with respect to set inclusion of vertex sets) circuit in G that is a connected subset of ( Z 2 , u ) . A simple closed curve in G with respect to u is called a Jordan curve (with respect to u ) if it separates the space ( Z 2 , u ) into exactly two components, i.e., if the subspace Z 2 J of ( Z 2 , u ) has exactly two components.

Let z = ( x , y ) Z 2 be a point. We put

H ( z ) = { ( x + i , y ) ; i { 1 , 0 , 1 } } , V ( z ) = { ( x , y + i ) ; i { 1 , 0 , 1 } } , D ( z ) = { ( x + i , y + i ) ; i { 1 , 0 , 1 } } , D ( z ) = { ( x + i , y i ) ; i { 1 , 0 , 1 } } .

Next, we put

A ( z ) = H ( z ) V ( z ) D ( z ) D ( z ) .

In the literature, the points of H ( z ) V ( z ) and A ( z ) different from z are said to be 4-adjacent and 8-adjacent to z , respectively. Here, for all z Z 2 , each of the sets H ( z ) , V ( z ) , D ( z ) , and D ( z ) will be called a basic segment. Note that basic segments may be regarded as digital (three-element) line segments (where H ( z ) is oriented horizontally, V ( z ) is oriented vertically, and D ( z ) and D ( z ) are oriented diagonally in Z 2 ).

For every point z Z 2 , we put

w { z } = H ( z ) if z = ( 4 k + 2 , y ) where k Z and y 4 l + 2 for every l Z , V ( z ) if z = ( x , 4 l + 2 ) where l Z and x 4 k + 2 for every k Z , A ( z ) if z = ( 4 k + 2 , 4 l + 2 ) , k , l Z , { z } otherwise

and, for every two-element subset { z , t } Z 2 , we put w { z , t } = w { z } w { t } z , t where

z , t = H ( z ) if z = ( 4 k + 2 + i , l ) and t = ( 4 k + 2 , l ) , k , l Z , i { 1 , 1 } , V ( z ) if z = ( k , 4 l + 2 + i ) and t = ( k , 4 l + 2 ) , k , l Z , i { 1 , 1 } , D ( z ) if z = ( 4 k + 2 + i , 4 l + 2 + i ) and t = ( 4 k + 2 , 4 l + 2 ) , k , l Z , i { 1 , 1 } , D ( z ) if z = ( 4 k + 2 + i , 4 l + 2 i ) and t = ( 4 k + 2 , 4 l + 2 ) , k , l Z , i { 1 , 1 } , { z , t } otherwise.

It was pointed out by one of the referees that z , t = { z , t , z + 2 ( t z ) } if one of the following three conditions is satisfied

1. z = ( 4 k + 2 , 4 l + 2 ) and t w ( z ) = A ( z ) ,

2. z = ( 4 k + 2 , 4 l ) and t w ( z ) = H ( z ) ,

3. z = ( 4 k , 4 l + 2 ) and t w ( z ) = V ( z )

or one of the three conditions obtained by interchanging z and t in (1)–(3) is satisfied while z , t = { z , t } otherwise.

Now, putting w A = { w B ; B A , card B < 3 } for every subset A X (card A > 2 ), we obtain an S 3 -closure operator on Z 2 . The closure operator w is demonstrated in Figure 1. For any point z Z 2 , a point u Z 2 , u z , belongs to w { z } if and only if there is an edge from z to u in the directed graph demonstrated in the left part of Figure 1. If { z , t } Z 2 is a two-element subset, then a point u Z 2 with u w { z } w { t } belongs to w { z , t } if and only if, in the directed graph demonstrated in the right part of Figure 1, z and t are the end points of a dotted line segment containing no other point of Z 2 (the dotted line segments are not edges of the graph), there is an edge from z or t to u , and the points z , t , and u lie on a line (with z or t lying between the other two points so that the set { z , t , u } is a basic segment with t w { z } or z w { t } – cf, the directed graph in the left part of the figure).

Clearly, any sequence ( ( x i , y i ) i 2 ) ( Z 2 ) 3 satisfying one of the following nine conditions is a w -connected element:

1. x 0 = x 1 = x 2 and y 0 = y 1 = y 2 ,

2. x 0 = x 1 = x 2 and there is k Z such that y i = 4 k + i for all i < 3 ,

3. x 0 = x 1 = x 2 and there is k Z such that y i = 4 k i for all i < 3 ,

4. y 0 = y 1 = y 2 and there is k Z such that x i = 4 k + i for all i < 3 ,

5. y 0 = y 1 = y 2 and there is k Z such that x i = 4 k i for all i < 3 ,

6. there is k Z such that x i = 4 k + i for all i < 3 and there is l Z such that y i = 4 l + i for all i < 3 ,

7. there is k Z such that x i = 4 k + i for all i < 3 and there is l Z such that y i = 4 l i for all i < 3 ,

8. there is k Z such that x i = 4 k i for all i < 3 and there is l Z such that y i = 4 l + i for all i < 3 , and

9. there is k Z such that x i = 4 k i for all i < 3 and there is l Z such that y i = 4 l i for all i < 3 .

It may easily be seen that, for any subset A Z 2 , z w A if and only if there exists a sequence ( ( x i , y i ) i 2 ) ( Z 2 ) 3 satisfying one of the above nine conditions such that z = ( x i 0 , y i 0 ) for some i 0 , 0 < i 0 2 , and ( x i , y i ) A for all i < 2 . Thus, w = w 2 so that w is a 2-discrete closure operator on Z 2 . The w -connected elements ( ( x i , y i ) i 2 ) ( Z 2 ) 3 satisfying (an arbitrary single) one of the conditions (2)–(9) are demonstrated in Figure 2 by directed line segments with the starting point ( x 0 , y 0 ) and the end point ( x 2 , y 2 ) (so that ( x 1 , y 1 ) is the midpoint of the segment).

We denote by H the graph with the vertex set Z 2 such that, for all z , t Z 2 , z and t are adjacent in H if and only if they are different and one of the following two conditions is satisfied:

1. z w { t } or t w { z } ,

2. there is a point u Z 2 , z u t , such that either z w { u } and t w { u , z } or t w { u } and z w { u , t } .

A graphical representation of (a section of) the graph H is given by Figure 2 when forgetting all edge directions.

The following results may be proved by applying Theorem 2.8 (in [17)], they were obtained by the help of Lemma 3.3 proved there, which is nothing but a straightforward consequence of Theorem 2.8):

1. The closure space ( Z 2 , w ) is connected.

2. Every circuit C in the graph H that turns only at some of the points ( 4 k , 4 l ) , k , l Z , is a Jordan curve in H with respect to the closure operator w and has the property that its union with any of the two components of the subspace Z 2 C of ( Z 2 , w ) is connected.

Jordan curves in the graph H with respect to the closure operator w will be briefly called Jordan curves in the closure space ( Z 2 , w ) . The possible turning points of the Jordan curves in ( Z 2 , w ) determined in statement (2) are the points ( 4 k , 4 l ) , k , l Z , where the curves may turn at the acute angle π 4 . This is an advantage over the Jordan curves with respect to the Khalimsky topology because they may never turn at the acute angle π 4 – cf. [16].

Figure 1

Closure operator w .

Figure 2

w -connected elements satisfying (an arbitrary but) one of the conditions (2)–(9).

3 Conclusion

We have introduced, for every ordinal α > 0 , closure operators that generalize the quasi-discrete closure operators [4]. These closure operators, called α -discrete, are studied. In particular, it is shown that the connectivity with respect to α -discrete closure operators is a certain type of path connectivity. This enables us to apply the operators to solving various problems of a discrete nature based on connectivity. One such application in digital topology is discussed. We have shown that there is a 2-discrete closure operator on the digital plane Z 2 allowing for a digital analogue of the Jordan curve theorem, thus providing a convenient structure on Z 2 for the study of digital images.

Acknowledgments

1. Funding information: This research was supported by the Brno University of Technology from the Specific Reasearch Project no. FSI-S-20-6187.

2. Author contributions: The author have accepted responsibility for the entire content of this manuscript and approved its submission.

3. Conflict of interest: The author states no conflict of interest.

4. Data availability statement: All data generated or analysed during this study are included in this published article.

References

[1] E. Čech, Topological spaces, in: Topological Papers of Eduard Čech, Academia, Prague, 1968, ch. 28, pp. 436–472. Search in Google Scholar

[2] G. Grätzer, General Lattice Theory, Birkhäuser Verlag, Basel, 1978. 10.1007/978-3-0348-7633-9Search in Google Scholar

[3] E. Giuli and J. Šlapal, Neighborhoods with respect to a categorical closure operator, Acta Math. Hungar. 124 (2009), 1–14. 10.1007/s10474-009-8108-zSearch in Google Scholar

[4] E. Čech, Topological Spaces (revised by Z.Frolík and M. Katětov), Academia, Prague, 1966. Search in Google Scholar

[5] T. Y. Kong, R. Kopperman, and P. R. Meyer, A topologicalapproach to digital topology, Amer. Math. Monthly 98 (1991), 902–917. 10.1080/00029890.1991.12000810Search in Google Scholar

[6] G. Chiaselotti, F. Infusino, and P. A. Oliverio, Set relations and set systems induced by somefamilies of integral domains, Adv. Math. 363 (2020), 106999. 10.1016/j.aim.2020.106999Search in Google Scholar

[7] G. Chiaselotti and F. Infusino, Alexandroff topologies and monoid actions, Forum Math. 32 (2020), 795–826. 10.1515/forum-2019-0283Search in Google Scholar

[8] S. Sarussi, Alexandroff topology of algebras over an integral domain, Mediterr. J. Math. 17 (2020), 54. 10.1007/s00009-020-1502-zSearch in Google Scholar

[9] X. Xu and D. Zhao, On topological Rudinas lemma, well-filtered spaces and sober spaces, Topology Appl. 272 (2020), 107080. 10.1016/j.topol.2020.107080Search in Google Scholar

[10] X. Xu, C. Shen, X. Xi, and D. Zhao, On T0-spaces determined by well-filtered spaces, Topology Appl. 282 (2020), 107323. 10.1016/j.topol.2020.107323Search in Google Scholar

[11] D. Zhao and W. K. Ho, On topologies defined by irreducible sets, J. Log. Algebr. Methods Program. 84 (2015), 185–195. 10.1016/j.jlamp.2014.10.003Search in Google Scholar

[12] G. Preuß, Allgemeine Topologie, 2nd ed., Springer-Verlag, Berlin, 1975. 10.1007/978-3-642-66211-9Search in Google Scholar

[13] M. B. Smyth, Semi-metrics, closure spaces and digital topology, Theoretic. Comput. Sci. 151 (1995), 257–276. 10.1016/0304-3975(95)00053-YSearch in Google Scholar

[14] A. Rosenfeld, Connectivity in digital pictures, J. Assoc. Comput. Mach. 17 (1970), 146–160. 10.1145/321556.321570Search in Google Scholar

[15] D. Marcus et al., A special topology for theintegers (Problem 5712), Amer. Math. Monthly 77 (1970), 1119. 10.2307/2316121Search in Google Scholar

[16] E. D. Khalimsky, R. Kopperman, and R. Meyer, Computer graphics and connected topologies onfinite ordered sets, Topology Appl. 36 (1990), 1–17. 10.1016/0166-8641(90)90031-VSearch in Google Scholar

[17] J. Šlapal, A closure operator for the digital plane, Filomat 34 (2020), 3229–3237. 10.2298/FIL2010229SSearch in Google Scholar