Connectivity Preserving Multivalued Functions in Digital Topology
aa r X i v : . [ c s . C V ] D ec Connectivity Preserving Multivalued Functions in Digital Topology
Laurence Boxer ∗ P. Christopher Staecker † Abstract
We study connectivity preserving multivalued func-tions [10] between digital images. This notion gener-alizes that of continuous multivalued functions [6, 7]studied mostly in the setting of the digital plane Z .We show that connectivity preserving multivaluedfunctions, like continuous multivalued functions, areappropriate models for digital morpholological opera-tions. Connectivity preservation, unlike continuity, ispreserved by compositions, and generalizes easily tohigher dimensions and arbitrary adjacency relations.Key words and phrases: digital topology, digitalimage, continuous multivalued function, shy map,morphological operators, retraction, simple point Continuous functions between digital images were in-troduced in [12] and have been explored in manysubsequent papers. However, the notion of a contin-uous function f between digital images X and Y doesnot always yield results analogous to what might beexpected from parallels with the Euclidean objectsmodeled by X and Y . For example, in Euclideanspace, if X is a square and Y is an arc such that Y ⊂ X , then Y is a continuous retract of X [1].However, [2] gives an example of a digital square X containing a digital arc Y such that Y is not a con-tinuous retract of X .In order to address such anomalies, digitally con- ∗ Department of Computer and Information Sciences, Nia-gara University, Niagara University, NY 14109, USA; and De-partment of Computer Science and Engineering, State Univer-sity of New York at Buffalo. E-mail: [email protected] † Department of Mathematics, Fairfield University, Fair-field, CT 06823-5195, USA. E-mail: cstaecker@fairfield.edu tinuous multivalued functions were introduced [6, 7].These papers showed that in some ways, digitallycontinuous multivalued functions allow the digitalworld to model the Euclidean world better than dig-itally continuous single-valued functions. However,digitally continuous multivalued functions have theirown anomalies, e.g., composition does not always pre-serve continuity among digitally continuous multival-ued functions [8].In this paper, we study connectivity preservingmultivalued functions between digital images andshow that these offer some advantages over continu-ous multivalued functions. One of these advantages isthat the composition of connectivity preserving mul-tivalued functions between digital images is connec-tivity preserving. Another advantage is that the con-cept of connectivity preservation of a map on a dig-ital image can be defined without any reference to aparticular realization of X as a subset of Z n ; by con-trast, an example discussed in Section 2 shows thatcontinuity of a multivalued map on ( X, κ ) is heavilyinfluenced by how X is embedded in Z n . These ad-vantages help us to generalize easily our definitionsand results to images of any dimension and adjacencyrelations.There are also disadvantages in the use of connec-tivity preserving multivalued functions as comparedwith the use of continous multivalued functions. Insection 7, we show ways in which continuous multival-ued functions better model retractions of Euclideantopology than do connectivity preserving multivaluedfunctions. We will assume familiarity with the topological the-ory of digital images. See, e.g., [2] for the standard1efinitions. All digital images X are assumed to carrytheir own adjacency relations (which may differ fromone image to another). When we wish to emphasizethe particular adjacency relation we write the imageas ( X, κ ), where κ represents the adjacency relation.Among the commonly used adjacencies are the c u -adjacencies. Let x, y ∈ Z n , x = y . Let u be aninteger, 1 ≤ u ≤ n . We say x and y are c u -adjacentif • There are at most u indices i for which | x i − y i | =1. • For all indices j such that | x j − y j | 6 = 1 we have x j = y j .We often label a c u -adjacency by the number ofpoints adjacent to a given point in Z n using this ad-jacency. E.g., • In Z , c -adjacency is 2-adjacency. • In Z , c -adjacency is 4-adjacency and c -adjacency is 8-adjacency. • In Z , c -adjacency is 6-adjacency, c -adjacencyis 18-adjacency, and c -adjacency is 26-adjacency.For much of the paper, we will not need to assumethat ( X, κ ) is embedded as a subset of ( Z n , κ ) forsome particular n .A subset Y of a digital image ( X, κ ) is κ -connected [12], or connected when κ is understood,if for every pair of points a, b ∈ Y there exists a se-quence { y i } mi =0 ⊂ Y such that a = y , b = y m , and y i and y i +1 are κ -adjacent for 0 ≤ i < m . The followinggeneralizes a definition of [12]. Definition 2.1. [3] Let (
X, κ ) and (
Y, λ ) be digitalimages. A function f : X → Y is ( κ, λ )-continuous iffor every κ -connected A ⊂ X we have that f ( A ) is a λ -connected subset of Y .When the adjacency relations are understood, wewill simply say that f is continuous . Continuity canbe reformulated in terms of adjacency of points: Theorem 2.2. [12, 3]
A function f : X → Y iscontinuous if and only if, for any adjacent points x, x ′ ∈ X , the points f ( x ) and f ( x ′ ) are equal oradjacent. For two subsets
A, B ⊂ X , we will say that A and B are adjacent when there exist points a ∈ A and b ∈ B such that a and b are equal or adjacent.Thus sets with nonempty intersection are automati-cally adjacent, while disjoint sets may or may not beadjacent. It is easy to see that a union of connectedadjacent sets is connected.A multivalued function f : X → Y assigns a subsetof Y to each point of x . We will write f : X ⊸ Y .For A ⊂ X and a multivalued function f : X ⊸ Y ,let f ( A ) = S x ∈ a f ( x ). Definition 2.3. [10] A multivalued function f : X ⊸ Y is connectivity preserving if f ( A ) ⊂ Y isconnected whenever A ⊂ X is connected.As is the case with Definition 2.1, we can reformu-late connectivity preservation in terms of adjacencies. Theorem 2.4.
A multivalued function f : X ⊸ Y is connectivity preserving if and only if the followingare satisfied: • For every x ∈ X , f ( x ) is a connected subset of Y . • For any adjacent points x, x ′ ∈ X , the sets f ( x ) and f ( x ′ ) are adjacent.Proof. First assume that f satisfies the two condi-tions above, let A be connected, and we will showthat f ( A ) is connected. Take two points y, y ′ ∈ f ( A ),and we will find a connected subset B ⊂ f ( A ) con-taining y and y ′ , and thus y and y ′ are connected bya path in f ( A ). Since y, y ′ ∈ f ( A ), there are points x, x ′ ∈ A with y ∈ f ( x ) and y ′ ∈ f ( x ′ ). Since A is connected there is a path x = x , x , . . . , x k = x ′ with x i ∈ A and x i adjacent to x i +1 for each i .By our hypotheses, we have f ( x i ) connected and f ( x i ) adjacent to f ( x i +1 ) for each i . Thus the union B = k [ i =0 f ( x i )2s connected, since it is a union of connected adjacentsets. So B ⊂ f ( A ) is connected and contains y and y ′ , which concludes the proof that f ( A ) is connected.Now for the converse assume that f is connectivitypreserving, and we will prove the two properties inthe statement of the theorem. The first property istrivially satisfied since f ( x ) = f ( { x } ) and { x } is con-nected. To prove the second property, assume that x, x ′ ∈ X are adjacent, and we will show that f ( x )and f ( x ′ ) are adjacent.Since x and x ′ are adjacent, the set { x, x ′ } is con-nected and thus the set f ( { x, x ′ } ) = f ( x ) ∪ f ( x ′ )is connected. Therefore, f ( x ) must be adjacent to f ( x ′ ).Definition 2.3 is related to a definition of multival-ued continuity for subsets of Z n given and exploredby Escribano, Giraldo, and Sastre in [6, 7] based onsubdivisions. (These papers make a small error withrespect to compositions, which is corrected in [8].)Their definitions are as follows: Definition 2.5.
For any positive integer r , the r -thsubdivision of Z n is Z nr = { ( z /r, . . . , z n /r ) | z i ∈ Z } . An adjacency relation κ on Z n naturally induces anadjacency relation (which we also call κ ) on Z nr as fol-lows: ( z /r, . . . , z n /r ) , ( z ′ /r, . . . , z ′ n /r ) are adjacentin Z nr if and only if ( z , . . . , z n ) and ( z , . . . , z n ) areadjacent in Z n .Given a digital image ( X, κ ) ⊂ ( Z n , κ ), the r -thsubdivision of X is S ( X, r ) = { ( x , . . . , x n ) ∈ Z nr | ( ⌊ x ⌋ , . . . , ⌊ x n ⌋ ) ∈ X } . Let E r : S ( X, r ) → X be the natural map sending( x , . . . , x n ) ∈ S ( X, r ) to ( ⌊ x ⌋ , . . . , ⌊ x n ⌋ ).For a digital image ( X, κ ) ⊂ ( Z n , κ ), a function f : S ( X, r ) → Y induces a multivalued function F : X ⊸ Y as follows: F ( x ) = [ x ′ ∈ E − r ( x ) { f ( x ′ ) } . A multivalued function F : X ⊸ Y is called continuous when there is some r such that F is X S ( X, Y S ( Y, X and Y with their secondsubdivisions.induced by some single valued continuous function f : S ( X, r ) → Y .An example of two spaces and their subdivisions isgiven in Figure 1.Note that the subdivision construction (and thusthe notion of continuity) depends on the particularembedding of X as a subset of Z n . In particularwe may have X, Y ⊂ Z n with X isomorphic to Y but S ( X, r ) not isomorphic to S ( Y, r ). This in factis the case for the two images in Figure 1, when weuse 8-adjacency for all images. The spaces X and Y in the figure are isomorphic, each being a set oftwo adjacent points. But S ( X,
2) and S ( Y,
2) arenot isomorphic since S ( X,
2) can be disconnected byremoving a single point, while this is impossible in S ( Y, X as being embedded inside of anyparticular integer lattice Z n . Proposition 2.6. [6, 7] Let F : X ⊸ Y be a con-tinuous multivalued function between digital images.Then • for all x ∈ X , F ( x ) is connected; and • for all connected subsets A of X , F ( A ) is con-nected. Theorem 2.7.
For ( X, κ ) ⊂ ( Z n , κ ) , if F : X ⊸ Y is a continuous multivalued function, then F isconnectivity preserving.Proof. By Proposition 2.6, for all connected subsets A of X , F ( A ) is connected. The assertion followsfrom Definition 2.3.3he subdivision machinery often makes it difficultto prove that a given multivalued function is contin-uous. By contrast, many maps can easily be shownto be connectivity preserving. Proposition 2.8.
Let X and Y be digital images.Suppose Y is connected. Then the multivalued func-tion f : X ⊸ Y defined by f ( x ) = Y for all x ∈ X isconnectivity preserving.Proof. This follows easily from Definition 2.3.
Proposition 2.9.
Let F : ( X, κ ) ⊸ ( Y, λ ) be a multivalued surjection between digital images ( X, κ ) , ( Y, κ ) ⊂ ( Z n , κ ) . If X is finite and Y is infi-nite, then F is not continuous.Proof. Since F is a surjection, X is finite, and Y isinfinite, there exists x ′ ∈ X such that F ( x ′ ) is aninfinite set. Therefore, no continuous single-valuedfunction f : S ( X, r ) → Y induces F , since for such afunction, S x ∈ E − r ( x ′ ) { f ( x ) } is finite. Corollary 2.10.
Let F : X ⊸ Y be the multivaluedfunction between digital images defined by F ( x ) = Y for all x ∈ X . If X is finite and Y is infinite andconnected, then F is connectivity preserving but notcontinuous.Proof. This follows from Propositions 2.8 and 2.9.Examples of connectivity preserving but not con-tinuous multivalued functions on finite spaces areharder to construct, since one must show that a givenconnectivity preserving map X ⊸ Y cannot be in-duced by any map on any subdivision. After somemore development we will give such an example inExample 7.6.Other terminology we use includes the following.Given a digital image ( X, κ ) ⊂ Z n and x ∈ X , the setof points adjacent to x ∈ Z n , the neighborhood of x in Z n , and the boundary of X in Z n are, respectively, N κ ( x ) = { y ∈ Z n | y is κ -adjacent to x } ,N ∗ κ ( x ) = N κ ( x ) ∪ { x } , and δ κ ( X ) = { y ∈ X | N κ ( y ) \ X = ∅} . Other notions of continuity have been given for mul-tivalued functions between graphs (equivalently, be-tween digital images). We have the following.
Definition 3.1. [14] Let F : X ⊸ Y be a multival-ued function between digital images. • F has weak continuity if for each pair of adjacent x, y ∈ X , f ( x ) and f ( y ) are adjacent subsets of Y . • F has strong continuity if for each pair of ad-jacent x, y ∈ X , every point of f ( x ) is adja-cent or equal to some point of f ( y ) and everypoint of f ( y ) is adjacent or equal to some pointof f ( x ). Proposition 3.2.
Let F : X ⊸ Y be a multivaluedfunction between digital images. Then F is connec-tivity preserving if and only if F has weak continuityand for all x ∈ X , F ( x ) is connected.Proof. This follows from Theorem 2.4.
Example 3.3. If F : [0 , Z ⊸ [0 , Z is defined by F (0) = { , } , F (1) = { } , then F has both weakand strong continuity. Thus a multivalued functionthat has weak or strong continuity need not have con-nected point-images. By Theorem 2.4 and Proposi-tion 2.6 it follows that neither having weak continuitynor having strong continuity implies that a multival-ued function is connectivity preserving or continuous. (cid:3) Example 3.4.
Let F : [0 , Z ⊸ [0 , Z be definedby F (0) = { , } , F (1) = { } . Then F is continuousand has weak continuity but does not have strongcontinuity. (cid:3) Proposition 3.5.
Let F : X ⊸ Y be a multivaluedfunction between digital images. If F has strong con-tinuity and for each x ∈ X , F ( x ) is connected, then F is connectivity preserving. roof. The assertion follows from Definition 3.1 andTheorem 2.4. Alternately, it follows from Proposi-tion 3.2, since strong continuity implies weak conti-nuity.The following shows that not requiring the imagesof points to be connected yields topologically unsat-isfying consequences for weak and strong continuity.
Example 3.6.
Let X and Y be nonempty digitalimages. Let the multivalued function f : X ⊸ Y bedefined by f ( x ) = Y for all x ∈ X . • f has both weak and strong continuity. • f is connectivity preserving if and only if Y isconnected. Proof.
That f has both weak and strong continuityis clear from Definition 3.1.Suppose f is connectivity preserving. Then for x ∈ X , f ( x ) = Y is connected. Conversely, if Y isconnected, it follows easily from Definition 2.3 that f is connectivity preserving.As a specific example consider X = { } ⊂ Z and Y = { , } , all with c adjacency. Then the function F : X ⊸ Y with F (0) = Y has both weak andstrong continuity, even though it maps a connectedimage surjectively onto a disconnected image. Connectivity preservation of multivalued functions ispreserved by compositions. For two multivalued func-tions f : X ⊸ Y and g : Y ⊸ Z , let g ◦ f : X ⊸ Z be defined by g ◦ f ( x ) = g ( f ( x )) = [ y ∈ f ( x ) g ( y ) . Theorem 4.1. If f : X ⊸ Y and g : Y ⊸ Z areconnectivity preserving, then g ◦ f : X ⊸ Z is con-nectivity preserving.Proof. We must show that g ◦ f ( A ) = g ( f ( A )) is con-nected whenever A is connected. Since f is connec-tivity preserving we have f ( A ) connected, and thensince g is connectivity preserving we have g ( f ( A ))connected. By contrast with Theorem 4.1, Remark 4 of [8]shows that composition does not always preserve con-tinuity in multivalued functions between digital im-ages. The example given there has finite digital im-ages X, Y, Z in Z and multivalued functions F : X → Y , G : Y → Z such that F is (4 , k )-continuousand G is ( k, k ′ )-continuous for { k, k ′ } ⊂ { , } , but G ◦ F : X → Z is not (4 , k ′ )-continuous. In fact,the example presented in [8] shows that even if F is a single-valued isomorphism, G ◦ F need not be acontinuous multivalued function. However, by The-orems 2.7 and 4.1, G ◦ F is (4 , k ′ )-connectivity pre-serving. Definition 5.1. [4] Let f : X → Y be a continuoussurjection of digital images. We say f is shy if • for each y ∈ Y , f − ( y ) is connected, and • for every y , y ∈ Y such that y and y areadjacent, f − ( { y , y } ) is connected.Shy maps induce surjections on fundamentalgroups [4]. Some relationships between shy maps f and their inverses f − as multivalued functions werestudied in [5], including a restricted analog of Theo-rem 5.2 below. We have the following. Theorem 5.2.
Let f : X → Y be a continuous sur-jection between digital images. Then f is shy if andonly if f − : Y ⊸ X is a connectivity preservingmultivalued function.Proof. This follows immediately from Theorem 2.4and Definition 5.1.
In [6, 7], it was shown that several fundamental oper-ations of mathematical morphology can be performedby using continuous multivalued functions on digitalimages. In this section, we obtain similar results us-ing connectivity preserving multivalued functions. Inorder to define the morphological operators, we must5ssume in this section that all images X under con-sideration are embedded in Z n for some n with aglobally defined adjacency relation κ . Thus in thissection we always have ( X, κ ) ⊂ ( Z n , κ ). The workin [6, 7] focuses exclusively on n = 2, and κ being4- or 8-adjacency. Our results have the advantageof being applicable in any dimensions and using any(globally defined) adjacency relation. In the following, the use of k = 4 or k = 8 indicates4-adjacency or 8-adjacency, respectively, in Z .Dilation [13] of a binary image can be regarded asa method of magnifying or swelling the image. Acommon method of performing a dilation of a digitalimage ( X, κ ) ⊂ ( Z n , κ ) is to take the dilation D κ ( X ) = [ x ∈ X N ∗ κ ( x ) . Theorem 6.1. ([7]; proof corrected in [8]) Given(
X, k ) ⊂ ( Z , k ), the multivalued functions ˜ D k : X → D k ( X ) ⊂ Z defined by ˜ D k ( x ) = N ∗ k ( x ), where k ∈ { , } , are both (4 , , Theorem 6.2.
Given a digital image ( X, κ ) ⊂ ( Z n , κ ) , the multivalued function ˜ D κ : X → D κ ( X ) ⊂ Z n defined by ˜ D κ ( x ) = N ∗ κ ( x ) is connectivity preserv-ing.Proof. For every x ∈ X , ˜ D κ ( x ) is κ -connected. Given κ -adjacent points x, x ′ ∈ X , we have x ′ ∈ ˜ D κ ( x ),so ˜ D κ ( x ) and ˜ D κ ( x ′ ) are κ -adjacent. The assertionfollows from Theorem 2.4.More general dilations are defined as follows. Let X ⊂ Z n be a digital image and let B ⊂ Z n , with theorigin of Z n a member of B . We call B a structuringelement . Given x ∈ Z n , let t x be the translation by x : t x ( y ) = x + y for all y ∈ Z n . The dilation of X by B is D B ( X ) = [ x ∈ X t x ( B ) . We have the following.
Theorem 6.3.
Let X ⊂ Z n be a digital image with c u -adjacency for ≤ u ≤ n and let B ⊂ Z n be astructuring element. If B is c u -connected, then themultivalued dilation function ˜ D B : X ⊸ D B ( X ) de-fined by ˜ D B ( x ) = t x ( B ) is connectivity preserving.Proof. Since B is c u -connected and t x is continuous,˜ D B ( x ) is connected for all x ∈ X . If x and x are c u -adjacent members of X and b ∈ B , then x + b and x + b are c u -adjacent, so ˜ D B ( x ) and ˜ D B ( x ) are c u -adjacent. The assertion follows from Theorem 2.4.Note that Theorem 6.3 is easily generalized to anyadjacency that is preserved by translations.There are non-equivalent definitions of the erosionoperation in the literature. We will use the definitionof [7]: the κ -erosion of X ⊂ Z n is E κ ( X ) = Z n \ D κ ( Z n \ X ) . In [7], we find the following.The erosion operation cannot be adequatelymodeled as a digitally continuous multival-ued function on the set of black pixels sinceit can transform a connected set into a dis-connected set, or even delete it (for exam-ple, the erosion of a curve is the empty setand, in general, the erosion of two discs con-nected by a curve would be the disconnectedunion of two smaller discs). However, sincethe erosion of a set agrees with the dila-tion of its complement, the erosion operatorcan be modeled by a continuous multivaluedfunction on the set of white pixels.It follows from Theorem 6.2 that the erosion op-erator can be modeled by a connectivity preservingmultivalued function on the set of white pixels. I.e.,as an analog of Corollary 6.4 below, we have Corol-lary 6.5 below. We use the notation E κ to suggestthat the function’s image is the compliment of theerosion. Corollary 6.4. ([7]; proof corrected in [8]) Given X ⊂ Z n , the multivalued function E k : Z \ X → Z given by E k ( y ) = N ∗ k ( y ) for y ∈ Z \ X is both (4 , , k ∈ { , } .6 orollary 6.5. Given ( X, κ ) ⊂ ( Z n , κ ) , the multi-valued function E κ : Z n \ X → Z n given by E κ ( x ) = N ∗ κ ( x ) is connectivity preserving.Proof. The assertion follows as in the proof of Theo-rem 6.2.
Like dilation, closing (or computing the closure of) adigital image can be regarded as a way to swell theimage.The closure operator C κ is the result of a dilationfollowed by an erosion. Since we have defined an ero-sion on X as a dilation on Z n \ X , we cannot saythat C κ is a composition of a dilation and an ero-sion, since the corresponding composition E κ ◦ ˜ D κ isnot generally defined. However, from the definitionsabove, the closure of X can be defined as C κ ( X ) = Z n \ ˜ D κ ( Z n \ [ x ∈ X N ∗ κ ( x )) . This yields the following results.
Theorem 6.6. [7] Given X ⊂ Z , the closure oper-ator C k is ( k, k )-continuous, k ∈ { , } . Theorem 6.7.
Given a digital image ( X, κ ) ⊂ ( Z n , κ ) , the closure operator C κ is connectivity pre-serving.Proof. Note we can define a multivalued function ˜ C κ : X ⊸ C κ ( X ) by˜ C κ ( x ) = (cid:26) { x } if x ∈ X \ δ κ ( X ); N ∗ κ ( x ) ∩ C κ ( x ) if x ∈ δ κ ( X ) . Since X ⊂ C κ ( X ) and each point of N ∗ κ ( x ) is κ -adjacent or equal to x , it follows that ˜ C κ ( x ) isconnected for all x ∈ X . Further, for κ -adjacent x, x ′ ∈ X , we have x ∈ ˜ C κ ( x ) and x ′ ∈ ˜ C κ ( x ′ ), so f ( x ) and f ( x ′ ) are adjacent. The assertion followsfrom Theorem 2.4.We find in [7] the following.As it happens in the case of the erosion, theopening operation (erosion composed with dilation) cannot be adequately modeled asa digitally continuous multivalued functionon the set of black pixels (the same exam-ples used for the erosion also work for theopening). However, since the opening of aset agrees with the closing of its complement[13], the k-opening operator can be modeledby a k-continuous multivalued function onthe set of white pixels.Thus, we define an opening operator for X as theclosure operator on Z n \ X . Corresponding to Corol-lary 6.8 below, we have Corollary 6.9 below. Corollary 6.8. [7] Given X ⊂ Z , the k -openingoperation on X can be modeled as a (4 , , O k : Z \ X → Z . Corollary 6.9.
Given ( X, κ ) ⊂ ( Z n , κ ) , the κ -opening operation on X can be modeled as a connec-tivity preserving function O κ : Z n \ X → Z n .Proof. The assertion follows from Theorem 6.7.
A continuous single-valued or multivalued function,or a connectivity preserving multivalued function, r ,from a set X to a subset Y of X is called a retraction [1], a multivalued retraction , or a connectivity preserv-ing multivalued retraction , respectively, if r ( y ) = y (respectively, r ( y ) = { y } ) for all y ∈ Y . In this casewe say Y is a retract of X , a multivalued retract of X , or a connectivity preserving multivalued retract of X , respectively. It is known [2] that the boundaryof a digital square is not a retract of the square. Bycontrast, we have the following. Example 7.1.
Let X = [ − , Z . Let Y = X \{ (0 , } . Then ( Y,
8) is a connectivity preserving mul-tivalued retract of ( X, roof. It is easy to see that the multivalued function r : X ⊸ Y given by r ( x ) = (cid:26) Y if x = (0 , { x } if x ∈ Y, is a connectivity preserving multivalued retraction of( X,
8) onto ( Y, Y,
8) is nota multivalued retract of ( X, r is connec-tivity preserving but not continuous.We can generalize the example given above in thefollowing result. The existence of connectivity pre-serving multivalued retractions is easily formulatedin terms of connected images: Theorem 7.2.
Let X be connected and let A ⊂ X , A = ∅ . Then A is a connectivity preserving multival-ued retract of X if and only if A is connected.Proof. First assume that A is connected. Then define f : X ⊸ A by: f ( x ) = ( { x } if x ∈ A,A if x A.f clearly has the retraction property that f ( A ) = A and f ( x ) = { x } for all x ∈ A . To show connectivitypreservation, let B ⊂ X be a connected set, and wewill show that f ( B ) is connected. In the case that B ⊂ A we have f ( B ) = B is connected. Otherwise, B \ A = ∅ so we have f ( B ) = A which was assumedto be connected. Thus f is connectivity preserving,so A is a connectivity preserving multivalued retractof X as desired.For the converse, assume that A is a connectiv-ity preserving multivalued retract of X . Since X isconnected, A must be connected.Theorem 7.2 makes it easy to tell when one set isa connectivity preserving multivalued retract of an-other. The analogous question for continuous multi-valued retracts is addressed in [7] (corrected in [8]),where the results are quite a bit more complicated,stated in terms of simple points , characterized by thefollowing. Definition 7.3. [9] Let X ⊂ Z . Let { k, k } = { , } .Let p ∈ X . Then p is a k -boundary point of X if andonly if N k ( p ) \ X = ∅ . (cid:3) Theorem 7.4. [11]
Let X ⊂ Z . Then p ∈ X is k -simple, k ∈ { , } , if and only if p is a k -boundarypoint of X and the number of k -connected compo-nents of N ( p ) ∩ X that are k -adjacent to p is equalto 1. Continuous multivalued retracts relate to simplepoints as follows:
Theorem 7.5. [8, Theorem 5] Let ( X, ⊂ Z be aconnected digital image, and let p ∈ X . Then X −{ p } is a continuous multivalued retract of X if and onlyif p is a simple point.The requirement that p be a simple point is astronger condition than X −{ p } being connected, thecondition for our Theorem 7.2. The authors of [8]also obtain a similar result for 4-adjacency requiringadditional hypotheses, and discuss removal of pairsof simple points. Their arguments become quite dif-ficult and do not seem able to address removal ofarbitrary subsets as in Theorem 7.2.Contrasting the results of Theorems 7.2 and 7.5gives examples of maps on finite spaces that are con-nectivity preserving but not continuous. In particu-lar, we have the following. Example 7.6.
Let X and Y be the images in Ex-ample 7.1. • The point (0 ,
0) is not a simple point of X andthus, Y is not a continuous multivalued retractof X , although Y is a connectivity preservingmultivalued retract of X . • The multivalued function r of Example 7.1 isconnectivity preserving but not continuous. Proof.
We saw in Example 7.1 that r is connectivitypreserving and that Y is a connectivity preservingmultivalued retract of X . • Clearly, (0 ,
0) is not a simple point of X . FromTheorem 7.5, Y is not a continuous multivaluedretract of X .8 Were r continuous then r would be a multivaluedretraction, contrary to Theorem 7.5. We have studied connectivity preserving multivaluedfunctions between digital images. This notion gen-eralizes continuous multivalued functions. We haveshown that composition, which does not preservecontinuity for continuous multivalued functions, pre-serves connectivity preservation for multivalued func-tions between digital images. We have obtained anumber of results for connectivity preserving multi-valued functions between digital images, concerningweak and strong continuity, shy maps, morphologi-cal operators, and retractions; many of our resultsare suggested by analogues for continuous multival-ued functions in [6, 7, 8, 5].
We are grateful for the suggestions of the anonymousreviewers.
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