Closure operations for digital topology-70996

Closure operations for digital topology

Closure operations

A set is closed (under AN operation) if and as long as the operation on any 2 components of the set produces another component of constant set. If the operation produces even one component outside of the set, the operation isn’t closed.

Consider the subsequent situations:

1. The set of real numbers is closed underneath addition. If you add 2 real numbers, you may get another complex quantity. there’s no risk of ever obtaining something apart from another complex quantity

2.         The set of integers isn’t closed underneath division

Since 2.5 isn’t a number, closure fails. There are different examples that fail.

3.         The set of real numbers is closed underneath multiplication. If you multiply 2 real numbers, you may get another complex quantity. There’s no risk of ever obtaining something apart from another complex quantity

4.         The set of real numbers isn’t closed underneath division

Since “undefined” isn’t a true range, closure fails.

Division by zero is that the solely case wherever closure fails for real numbers.

5.         A binary table of values is closed if the weather within the table area unit restricted to the weather of the set

This table shows operation outlined on the set. This operation is closed on this set since the weather within the table area unit restricted to solely the weather within the set.

Digital topology could be a theory that arose within the late Sixties for the study of geometric and topological properties of digital pictures. It well-tried to be a great tool for determination some problems of computer graphics and image process. Despite its name, the speculation was originally supported utilizing graph-theoretic instead of topological strategies.

 It was solely within the late Nineteen Eighties that a topological approach to digital topology was used for the, rest time in [3]. The most purpose of this approach is to supply the set Z of

Integers with a topological structure causation a product structure on Z×Z suitable for

Digital image process. Up to date, a series of papers are printed that develop the topological approach to digital topology. Most of them use the so-called Khalimsky topology on Z. during this note we have a tendency to generalize the topological approach based on the employment of the Khalimsky topology. we have a tendency to begin with learning closure operations on capricious sets so limit our issues to Z—we introduce a sequence  of suitable closure operations on Z whose ,rest member coincides with the Khalimsky topology. Thus, the results obtained by investigation these closure operations generalize some results well-tried for the Khalimsky topology in [5].

All closure operations studied area unit related to and this reality is employed with advance in our topological investigation.

Example 1. Let X = {a, b, c, d} and R = {(a, a), (a, b), (b, d), (c, d), (d, a)} be a relation on X, then

S1 = {{a, b}, {d}, {a}},

S2 = {{a, d}, {a}, {b, c}},

β1 = {{a}, {d}, {a, d}, φ, X}, and

β2 = {{a}, {a, d}, {b, c}, φ, X}.

Note that X ∈ β, since by definition X is the empty intersection of members of S [8]. So that,

τ1 = {φ, X, {a}, {d}, {a, b}, {a, d}, {a, b, d}}

and

τ2 = {φ, X, {a}, {a, d}, {b, c}, {a, b, c}}.

We shall denote the complement of any subset A of X by Ac

Proposition 1. In a closure space (X, cl1) (resp. (X, cl2)) if R is a transitive relation on X, then A ∈ N(x) ⇔ Int1 (A) (resp.Int2(A)) ∈ N(x).

Proposition 2. In a closure space (X, cl1) (resp.(X, cl2)) if R is a symmetric relation on X, then cl1(A) (cl2(A)) is the minimal neighbourhood of a set A.

Proof. If R be a symmetric relation, then N1(A) = A ∪ {y ∈ X : ∃x ∈ A, xRy} =A ∪ {y ∈ X : ∃x ∈ A, yRx} = cl1(A). Moreover, we have N2(A) = A ∪ {y ∈ X :∃x ∈ A, yRx} = A ∪ {y ∈ X : ∃x ∈ A, xRy} = cl 2(A).

The important idea is that, we give another definition of the minimal neighborhood of a point x in a closure space (X, cl3) as follows:

N3(x) =(hxiR if hxiR 6 = φ,{x} if hxiR = φ.

Also, from Lemma  we get.

Proposition 3. In a closure space (X, cl3) A ∈ N(x) ⇔ Int 3(A) ∈ N(x) for any

binary relation R on X.

Lemma 2. In a closure space (X, cl3) the open sets are precisely the unions ∪x∈A(N3(x)) for all A ⊆ X.

Proof. Let A be an open set in (X, cl3), then A = Int3(A) = {x ∈ A : hxiR ⊆ A}.

Hence A is a neighborhood of each of its elements, so for each y ∈ A, we have

N3(y) ⊆ A, then ∪x∈A(N3(x)) ⊆ A. But since y ∈ N3(y) for all y ∈ X, we have A ⊆ ∪x∈A(N3(x)).

And so A is the union of the minimal neighborhoods of its elements. Conversely, consider any subset A ⊆ X. We want to show that ∪x∈A(N3(x)) is an open set. We will show that N3(x) is open. First if hxiR 6 = φ, then for any point y ∈ N3(x) = hxiR we have hyiR ⊆ hxiR, and hence y ∈ Int3(hxiR) = Int3(N3(x)), thus N3(x) is open.

Second if hxiR = φ then N3(x) = {x} = {x ∈ {x} : hxiR ⊆ {x}} = Int3({x}), i.e.,

N3(x) = Int3(N3(x)), then N3(x) is an open set.

Example

X is said to be connected in case X mainly not be able to represented in the form of two not number sets union which are Disjoint and open sets.

Given as

R³ be standard topology

X is subset of the R³

Hence

Clearly X cannot for be represented by union of two non empty Disjoint open sets.

Concludes X is connected

Concluding remarks

Here, we are going to shortly define the additional development of the speculation that we have a tendency to ideate and conceive to contribute to. Given a natural number n, by an array digital m-dimensional space we perceive the merchandise of m copies of the S∗n -space (Z;un). to point out that AN n-ary digital m-dimensional area is beneficial for determination issues of digital topology, we have to verify that it behaves, in a very bound sense, just like the real m-dimensional (topological) space. As so much as applications area unit involved, the foremost necessary cases area unit m= two and 3, i.e., then-ary digital plane and therefore the n-ary digital three-d area. The binary digital plane has been investigated in several papers  and there are some papers dedicated to the study of the binary digital three-d area

To develop our theory, it’s natural to begin with investigation then-ary digital plane

analogy is well-tried in [3]). this might alter North American country to switch the sometimes used Khalimsky topology v2 onZ2 with a closure operation vn wheren¿2 could be a number. To justify such a replacement, we’d like to point out that, for n¿2,vn has some benefits over

v2. one in every of the benefits may end up from the very fact that for n¿2 the closure operations vn aren’t any additional quasi-discrete.

Added in proofs: AN analogy of the simple closed curve theorem has been well-tried for

the n-ary digital planes. A simple closed curve theorem with respect to bound closure operations on the digital plane, Electronic Notes in Theoretical Computer Science forty six (2001)].

References

1. L. A. Ankeney and G. H. Ritter  “Cellular Topology and its Applications in Image Processing”,  Int. Journ. Compt. Inf. Sciences,  vol. 12,  pp.433 -455 1983
2. B. Bollobás  Graduate Texts in Math.,  vol. 63,  1979 :Springer-Verlag
3. E. Khalimsky , R. Kopperman and P. R. Meyer  “Computer graphics and Connected Topologies on finite ordered sets”,  Topology and Appl.,  vol. 36,  pp.1 -17 1990
4. T. Î¥. Kong and A. W. Roscoe  “A Theory of Binary Digital Pictures”,  Comput. Vision Graphics Image Process.,  vol. 32,  pp.221 -243 1985
5. T. Y. Kong and A. Rosenfeld  “Digital Topology: Introduction and Survey”,  Computer Vision, Graphics and Image Processing,  vol. 48,  pp.357 -393 1989
6. R. Kopperman , P. R. Meyer and R. G. Wilson  “A Jordan Surface Theorem for three-dimensional Digital Spaces”,  Discrete and Computational Geometry,  vol. 6,  pp.155 -161 1991
7. E. Kovalevsky  “The topology of cellular complexes as applied to image processing”,  Proc. II Int. Conference CAIP\’87 on Automatic Image Processing,  pp.162 -173 1987
8. S. Winter, “Bridging Vector and Raster Representation in GIS”, In R. Laurini, K. Makki, and N. Pissinou, (Eds.), Proceedings of the 6th International Symposium on Advances in Geographic Information Systems, pp. 57-62, Washington, D.C., ACM Press, 1998.
9. G. I. Metternicht, “Categorical fuzziness: a comparison between crisp and fuzzy class boundary modelling for mapping salt-affected soils using Landsat TM data and a classification based on anion ratios”, Elsevier Science, Vol. 168, 2003.
10. G. Bragato, “Fuzzy continuous classification and spatial interpolation in conventional soil survey for soil mapping of the lower Piave plain”, Elsevier Science, Vol. 118, 2004.
11. C. Maire and M. Datcu, “Object and topology extraction from remote sensing images”, Image Processing 2005, ICIP 2005, IEEE International Conference on, Vol. 2, Iss., pp: II-193-6, 2005.
12. P.-L. Bazin , L. Ellingsen and D. Pham  “Digital homeomorphisms in deformable registration”,  Proc. Int. Conf. Information Processing in Medical Imaging 2007(IPMI\’07),  vol. LNCS 4584,  pp.211 -222
13. J.-F. Mangin , V. Frouin , I. Bloch , J. Regis and J. L. pez Krahe  “From 3-D magnetic resonance images to structural representations of the cortex topography using topology preserving deformations”,  J. Math. Imaging Vis.,  vol. 5,  no. 4,  pp.297 -318 1995
14. N. Passat , C. Ronse , J. Baruthio , J. Armspach , C. Maillot and C. Jahn  “Region-growing segmentation of brain vessels: An atlas-based automatic approach”,  J. Magn. Reson. Imaging,  vol. 21,  no. 6,  pp.715 -725 2005
15. Y. Zeng , D. Samaras , W. Chen and Q. Peng  “Topology cuts: A novel min-cut/ max-flow algorithm for topology preserving segmentation in n-d images”,  Comput. Vis. Image Underst.,  vol. 112,  no. 1,  pp.81 -90 2008
16. F. Segonne , J. Pacheco and B. Fischl  “Geometrically accurate topology-correction of cortical surfaces using nonseparating loops”,  IEEE Trans. Med. Imaging,  vol. 26,  no. 4,  pp.518 -529 2007
17. B. Fischl , A. Liu and A. Dale  “Automated manifold surgery: Constructing geometrically accurate and topologically correct models of the human cerebral cortex”,  IEEE Trans. Med. Imaging,  vol. 20,  no. 1,  pp.70 -80 2001
18. M. Amu, K. MISCHAKOW AND A. TANNENBAUM. Cubical homology and the topological classification of 2D and 3D imagery. In “Proc. IEEE Internat. Conf. Image Process., 2001”, 173-176.
19. P. BENDICH, H. EDELSBRUNNER AND M. KERBER. Computing robustness and persistence for images. IEEE Trans. Visual. Comput. Graphics 16 (2010), 1251-1260.
20. P. BENDICH, H. EDELSBRUNNER, D. MOROZOV AND A. PATEL. Homology and robustness of level and interlevel sets. Homology, Homotopy, and Applications, to appear.
21. C. CHEN AND M. KERBER. An output-sensitive algorithm for persistent homology. In “Proc. 27th Ann. Sympos. Comput. Geometry, 2011”, 207-215.
22. D. COHEN-STEINER, H. EDELSBRUNNER AND J. HARER. Extending persistence with Poincaré and Lefschetz duality. Found. Comput. Math, 9 (2009), 79-103.
23. A. IYER-PASCUZZI, O. SYMONOVA, Y. MILEYKO, Y. HAO, H. BELCHER, J. HARER, J. S. WETTZ AND P. BENFEY. Imaging and analysis platforms for automatic phenotyping and classification of plant root systems. Plant Physiol. 152 (2010), 1148-1157.
24. E. KHAUMSKY, R. KOPPERMAN AND P. R. MEYER. Computer graphics and connected topologies on finite ordered sets. Topology Appl. 36 (1990), 1-17.
25. N. PASSAT, M. COUPRIE, L. MAZO AND G. BERTRAND. Minimal simple sets: a new concept for topology-preserving transformations. In “Proc. 1st Workshop Combin. Top. Image Context, 2008”.

Topological structuring of the digital plane

In the classical approach to digital topology graph a priori tools area unit used for structuring Z2 , specifically the well-known binary relations of 4-adjacency and 8-adjacency. However neither 4-adjacency nor 8-adjacency itself permits AN analogue of the simple closed curve theorem (cf. [9]) and, therefore, one has got to use a mixture of the 2 adjacencies. to beat this disadvantage, a new, strictly topological approach to the matter was projected in [6] that utilizes a convenient topology on Z2, referred to as the Khalimsky topology (cf. [5]), for structuring the digital plane. At present, this topology is one in every of the foremost necessary ideas of digital topology. It’s been studied and utilized by several authors, see e.g.

[3] And [7]-[10].

The possibility of using convenient topological structures on Z2different from the Khalimsky topology is mentioned in [14]-[19]. Notably, in [16], a replacement topology on Z2

is introduced and it’s shown there that this topology provides bound convenient Jordan curves behaving additional well than the Jordan curves within the Khalimsky area. The quotient topologies of the topology from [16] area unit studied in [17] wherever it’s shown that they embrace, among others, the Khalimsky and Marcus-Wyse topologies. Within the gift note we have a tendency to continue the investigations from [16] and [17]. We have a tendency to discuss a topology on Z2 that is finer than the topology introduced in [16] however still has the property that the Khalimsky and Marcus-Wyse topologies belong to its quotient topologies. We have a tendency to study another of its quotient topologies on Z2, denoted by v, and prove a simple closed curve theorem for it. This simple closed curve theorem differs from the simple closed curve theorems for the geometer plane, the Khalimsky plane, and the (4, 8) and (8, 4) digital planes of”classical” digital topology within the following vital ways:

(i) The paper’s simple closed curve theorem solely applies to bound si to bound easy closed curves in (Z2, v). There are a unit every which way long easy closed curves C in (Z2, v) that Z2 has over 2 elements.

Jordan’s Curve Theorem

A very basic property — for theory in addition as for applications — of the topology of the plane is that Jordan’s Curve Theorem is valid. Meaning that any “simple closed curve” has the property of separating the plane into 2 components, particularly the inside with relevance the curve and also the exterior. These 2 components of the plane area unit distinguished by the actual fact that the latter isn’t delimited. For our functions we have a tendency to solely want this theorem for two-dimensional figure curves. A two-dimensional figure curve or just curve within the plane consists of a finite range of points referred to as vertices such every 2 consecutive vertices xi, xi+1 area unit joined by a line section referred to as a grip. A two-dimensional figure curve is termed a straightforward (polygonal) curve if edges meet solely in vertices and if for every vertex there are a unit at the most 2 edges meeting it. it’s termed a closed (polygonal) curve if x0 = xn.

Since Z2 will be thought-about as a set of R2, any path in Z2 corresponds to a two-dimensional figure curve within the plane R2 with vertices in Z2 and each arc corresponds to a straightforward curve within the plane. Similarly, a losed path (arc) corresponds to a closed two-dimensional figure curve (simple curve).

Theorem one (Jordan’s Curve Theorem) Given a straightforward closed two-dimensional figure curve C within the plane R2. Then R2 consists of precisely 2 open connected sets (in the sense of the standard R2 –topology). precisely one in all these sets is delimited and is named the inside with relevance C and also the different one is infinite and is named the outside with relevance C. The proof of this Theorem is kind of elementary however somewhat long.

The ‘digital analogy’ of this basic Theorem will be naively developed within the following way: Given a closed easy curve P within the digital plane Z2.

Then Z2 consists of precisely 2 connected sets. precisely one in all these sets is delimited and is named the inside with relevance P and also the different is infinite and is named the outside with relevance P .

If we have a tendency to interpret the term ‘curve’ as ‘arc’ then the assertion of the concept as developed here isn’t true. the subsequent 2 sets area unit closed arcs within the sense of the definition higher than however the complement (with relevance Z2) of every of them consists of only 1 connected element. Arrows within the figures indicate the order obligatory by list the points a minimum of four points and a closed digital 4–curve should have a minimum of eight points. Since there exist (up to translations, rotations by multiples of 90o and reflections at diagonal lines of the digital plane) solely finitely several arcs having fewer than four or eight points, severally, the reader will simply verify by inspecting all doable cases that this restriction so is sensible. within the following, a digital curve is known to be associate degree arc that fulfils the restriction mentioned higher than if it’s closed.

There is, however, a second drawback. contemplate the 2 following digital sets (black points (•) area unit understood to belong to S, white points (◦) to the complement):

The first set is associate degree 8–curve (in the 4–sense it’s not a curve). Its complement, however, consists of only 1 8–connected element. equally the second set may be a 4–curve (but not associate degree 8–curve) and its complement consists of 3 4–components. These phenomena area unit typically referred to as ‘connectivity paradoxa’.

In 1979 Rosenfeld proved that Jordan’s curve Theorem is so true for digital curves if the curve and its complement area unit equipped with totally different ‘topologies’. This was determined earlier by Duda,

Theorem two (Digital Jordan Theorem) Given a closed easy κ–curve P within the digital plane Z2.

Then Z2 consists of precisely 2 ¯ κ–connected sets. precisely one in all these sets is delimited and is named the inside with relevance P and also the different is infinite and is named the outside with relevance P . Rosenfeld’s proof of this Theorem is performed primarily by associate degree embedding approach. we have a tendency to come to the present subject afterward.

In image process applications the sets of black points area unit thought-about to hold the essential data of a black–white image. thus sometimes the set of all black points is provided with the 8–topology. since this topology exhibits a additional advanced property structure. we’ll follow this practice here and conjointly use the 8–topology. it’s simply doable to get assertions regarding the 4–topology by work the negative image that is obtained by dynamical the roles of S and its complement. This must be administered with some caution since the case isn’t quite radially symmetrical as a result of we have a tendency to typically assume S to be delimited.

Digital curves play a special role in image process applications. initial it’s doable to represent digital curves during a storage economical method by storing the coordinates of P0 then solely the variations Pi+1−Pi for i = zero, 1, • • • , n − 1. These variations will be coded by storing solely the amount within the neighborhood configuration of Pi that corresponds to Pi+1. This code for a digital path is known as chain code. For cryptography eight neighbors of a degree one wants three bits, thus a curve of length n will be hold on exploitation 3n bits if one neglects the number of storage necessary to store the primary purpose P0. For storing binary pictures it’s comfortable to code solely the boundaries of the black digital sets. The justification of representing a digital set by its boundary is given by Jordan’s Curve Theorem. A binary image consisting of n × n points will be hold on exploitation n × n bits. If solely the boundaries of the black sets area unit hold on, one wants 3nB bits, wherever nB is that the range of boundary points. cryptography the boundaries pays once {the range the amount the quantity} of boundary points is a smaller amount than thirty three take advantage of the overall number of points within the image. In usual text documents the amount of boundary points amounts to but ten take advantage of the amount of all points, and this figure is even smaller in line-structured pictures like engineering drawings. With growing resolution of scanners the amount of boundary points grows roughly linear with the amount of discretization points per unit length, the overall range of image points (as well because the range of black points), however, grows because the sq. of the amount of discretization points per unit length. As a consequence, boundary cryptography of pictures becomes additional and additional enticing with growing resolution power of scanning devices.

 The graphs of 4– and 8–topologies

The approach conferred during this section to ‘topologize’ the digital plane is especially supported graph speculative concepts. It uses ideas like points of Z2 being the nodes of the graph, and a neigborhood relation that is pictured by the perimeters of the (undirected) graph.

This model enabled North American nation to talk regarding property of sets during a graph speculative manner. The graphs representing 4– and 8–connectivity area unit represented in Figure1.

The graph reminiscent of the 4–topology is placoid, which suggests that it will be drawn within the plane such the lines representing the neighborhood relation meet solely in vertices. In distinction, it’s unattainable to represent the property of 8–topology by suggests that of a placoid graph. this means that the notions “planarity” and “topology of the digital plane” aren’t equivalent.

In order to prove that the 8–topology can’t be modelled by a placoid graph we have a tendency to show that the Kuratowski graph K5 will be drawn into the graph of the digital plane equipped with the 8–topology. consistent with Kuratowski’s theorem, a graph isn’t placoid if and given that it’s a subgraph homeomorphic to K5 or K3,3 (see [22, Theorem 11.13]). K5 is that the complete graph having five nodes. In Figure two it’s shown the way to insert K5 homeomorphically into the graph reminiscent of the 8–topology.

Example 1

(X; T ) is defined as Hausdoroff  in the topological space if

such that

such that

Consider

Now

Hence

Now

Also because

Hence for condition

It the intersection is not null because points contradict hence it satisfies

It concludes that for each distinct points pairs in the (x,y) which are contain in X, there does not exists the corresponding disjoint neighbors of Vy and Vx of y and x in that order.

Hence X is NOT Hausdoroff

Example 2

X is Hausdorff: Let x, y  X. If x, y  C2, then {x}, {y} are the disjoint open sets we’re looking for. If x, y  Cl, then we may clearly choose distinct subbasis elements of the second type by taking a small enough ‘interval”. If x  Cl, say, and y  C2, then whether or not y is the radial projection of x, we can always choose a subbasis interval of the second kind which includes but not y, then take {y} as the second open set.

X is not connected: Since X is Hausdorff, single point sets are closed. Any single point set in C2 is also open. Thus we have a proper, nonempty subspace of X (in fact, uncountable many of them) which is both open and closed, and therefore X is not connected. (Actually, the connected components of X turn out to be Cl and every single point set.

X is compact: Take any cover of X, then it is also a cover of Cl. As previously noted, Cl in the subspace topology is just Sl, which is compact. Therefore, we can take a finite subcover of our original cover and cover Cl. Returning to X, we note that since any open set in X containing Cl can miss only a finite number of points of C2, so now we just throw in one open set containing each of them from the original cover.

X is not separable: Suppose D is dense in X. Then D = X. Let x  C2. Since every open set containing x must intersect D (because the closure of D is all of X), in particular, the single point set {x} intersects D, i.e., x D. Thus D contains all of C2 and must therefore be uncountable.

B is basic topology

X is Ts for if M and N are two separated sets then it implies  and  are separated and thus concludes with EucIidean neighbourhoods (disjoint)  and  in C1. Now if  has a neighbourhood Vm which is disjoint from N, and  which can be choosen such that  likewise for  we can say Vb such that  and  then since  is open,  and   are disjoint neighbours of M and N.

1.  That X is Hausdorff mav be obsrved by a simple thought of cases.
2. X is Ts for if M and N are two separated sets then M and  Nare separated and therefore have displace EucIidean neighbourhoods VM and VN in C1.  Now if  has a neighbourhood VM which is displace from N, and  that might be selected such that  Similarly for  we may find VN such that and  then since  is open,  and   are disjoint neighbours of M and N.
3. X is sequentially compact in the topology of type EucIidean such that whichever sequence satisfies the convergent sequence. This subsequence can be seen to converge if consider the case of the topology of concentric circle denoted by the T  with respect to the limit point for Euclidean were present in C1  so as to point might be limit point r while if the limit  point Euelidean  might be in C2 for limit point Euelidean r might be the corresponding C1 projection. Consequently (X,r) is sVNsequently compact
4. X cannot be said to be separable because no countable set in C2 is dense. Hence X is compact.

Example 3

Suppose that U, V is a separation of R (we will understand throughout that we mean R with the new topology). with no loss of generalization, we can take for granted that there exist    and v € V such that . Then we let p be the least upper bound of the set

If p € U, then since and since U is open, we can find a basis element Bp such that  and in any case we get a contradiction of p being the least upper bound of Uv.

If p V, as above we can find a basis element Bp such that . Suppose now that pX. Then we may assume that Bp is of the 2nd type, that is,  for some      a, b  R. We would like to contradict the fact that p is a least upper bound for Uv again. The danger is that the interval (a, b) may contain points of U arbitrarily close to p;

Suppose that z €- (a, b). Then if zU, since U is open, there exists a basis element of the fourth type containing z, contained entirely in U. However, such a basis element necessarily intersects Bp, and so z must be an element of V. Thus V must contain a basis element of the form Zed for every z  (a, b), which implies that  is also contained in V. so we have p  (a, b) C V, which clearly contradicts p being a least upper bound for

Hence proved

Refrences

1. Jean–Paul Auray, Gerard Duru and Michel Mougeot. Some pretopological properties of input–output models and graph theory. Oper.Res. Verfahren , 34:5–21, 1979
2.  K. H. Baik and L. L. Miller. Topological approach for testing equivalence in heterogeneous relational databases. The Computer Journal ,33:2–10, 1990.
3. H. P. Barendregt. The Lambda Calculus — Its Syntax and Semantics , Revised Edition, Elsevier Science Publishers B.V., North–Holland, Amsterdam, New York, Oxford, 1984.
4. Marcel Brissaud. Structures topologiques des espaces pr´ef´erenci´es. Comptes Rendus hebdomadaires des S´eances de l’Academie des Sciences , 280, S´erie A:961–964, 1975
5. Frank Harary. Graph Theory , Addison–Wesley Publishing Company, Reading, Massachusetts, Menlo Park, California, London, Don Mills, Ontario, 1969.
6. Gabor T. Herman. Discrete multidimensional Jordan surfaces. CVGIP: Graphical Models and Image Processing , 54:507–515, 1992.
7. Gabor T. Herman. On topology as applied to image analysis. Computer Vision, Graphics, and Image Processing , 52:409–415, 1990.
8. John L. Kelley. General Topology , (The University Series in Higher Mathematics), D. Van Nostrand Company, Inc., Princeton, New Jersey, Toronto, London, New York, 1955.
9. Efim Davidoviˇc Halimski˘ı. Ordered Topological Spaces , (Russian), Academy of Sciences of the Ukrainian SSR, Naukova Dumka, Kiev, 1977.
10.  E. Khalimsky. Topological structures in computer science. Journal of Applied Mathematics and Simulation , 1:25–40, 1987.
11. T. Y. Kong, A. W. Roscoe and A. Rosenfeld. Concepts of digital topology. Topology Appl., 46:219–262, 1992.
12.  T. Y. Kong and E. Khalimsky. Polyhedral analogs of locally finite topological spaces. In: R. M. Shortt, ed.: General Topology and Applications , Lecture Notes in Pure and Applied Mathematics, Volume 123 , pages 153–163. New York, Basel: Marcel Dekker, Inc. 1990.
13. T. Y. Kong and A. Rosenfeld. If we use 4– or 8–connectedness for both the objects and the background, the Euler characteristic is not locally computable. Pattern Recognition Letters , 11:231–232, 1990.
14. T. Y. Kong and A. Rosenfeld. Digital topology: Introduction and survey. Computer Vision, Graphics, and Image Processing , 48:357– 393, 1989.
15. T. Y. Kong. A digital fundamental group. Computers & Graphics , 13:159–166, 1989.
16. T. Y. Kong and A. W. Roscoe. Continuous analogs of axiomatized digital surfaces. Computer Vision, Graphics, and Image Processing , 29:60–86, 1985.
17. Ralph Kopperman, Paul R. Meyer and Richard G. Wilson. A Jordan surface theorem for three–dimensional digital spaces. Discrete Comput. Geom., 6:155–161, 1991.
18. Ralph Kopperman and Yung Kong. Using general topology in image processing. In: U. Eckhardt, A. H¨ubler, W. Nagel and G. Werner, editors: Geometrical Problems of Image Processing. Proceedings of the 5th Workshop held in Georgenthal, March 11–15, 1991. (Research in Informatics, Volume 4), pages 66–71. Berlin:Akademie–Verlag 1991.
19. V. A. Kovalevsky. Finite topology as applied to image analysis. Computer Vision, Graphics, and Image Processing , 45:141–161, 1989.
20.  E. H. Kronheimer. The topology of digital images. Topology and its Applications , 46:279–303, 1992.
21.  Longin Latecki, Ulrich Eckhardt and Azriel Rosenfeld. Well–Composed Sets. CVGIP: Image Understanding , (to appear).
22. Longin Latecki. Topological connectedness and 8–connectedness in digital pictures. CVGIP: Image Understanding , 57:261–262, 1993
23. Dana Scott. Outline of a mathematical theory of computation. In Proceedings of the Fourth Annual Princeton Conference on Information Sciences and Systems (1970), pages 169–176.
24. Jean Serra, editor. Image Analysis and Mathematical Morphology, Volume 2: Theoretical Advances , Academic Press, Harcourt Brace Jovanovich Publishers, London, San Diego, NewYork, Boston, Sydney, Tokyo, Toronto, 1988.
25. Jean Serra. Image Analysis and Mathematical Morphology, Academic Press, Inc., London, Orlando, San Diego, New York, Austin, Boston, Sydney, Tokyo, Toronto, 1982