Connectivity with respect to α-discrete closure operators

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 .

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 .

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 .

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 .□

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

Next, we put

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

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

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].