Properties of Space Set Topological Spaces

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1 Filomat :9 (), DOI.98/FIL975H Published by Faulty of Sienes and Mathematis, University of Niš, Serbia Available at: htt:// Proerties of Sae Set Toologial Saes Sang-Eon Han a a Deartment of Mathematis Eduation, Institute of Pure and Alied Mathematis Chonbuk National University, Jeonu-City Jeonbuk, 589, Reubli of Korea Abstrat. Sine a loally finite toologial struture lays an imortant role in the fields of ure and alied toology, the aer studies a seial kind of loally finite saes, so alled a sae set toology (for brevity, SST) and further, roves that an SST is an Alexandroff sae satisfying the searation axiom T. Unlike a oint set toology, sine eah element of an SST is a sae, the resent aer names the toology by the sae set toology. Besides, for a onneted toologial sae (X, T) with X = the axioms T, semi-t and T are roved to be equivalent to eah other. Furthermore, the aer shows that an SST an be used for studying both ontinuous and digital saes so that it lays a ruial role in both lassial and digital toology, ombinatorial, disrete and omutational geometry. In addition, a onneted SST an be a good examle showing that the searation axiom semi-t does not imly T.. Introdution Finite, loally finite and, more generally, Alexandroff saes (or A-saes for short) [] were investigated by many authors: Alexandroff, MCord, Stong, and more reently by May, Hardie and Vermeulen, Barmak and Minian. Further, both simliial and abstrat ell (for short, AC) omlexes in [] (see also [, 7, 8]) have been often used in studying ure and alied toology relevant to both omuter and disrete geometry as well as omuter imaginary. However, we now need to oint out that they an be transformed into toologial saes in terms of a olyhedron and an axiomati toologial method, whih are a Hausdorff sae and a omliated loally finite sae, resetively [7, 8]. Besides, the earlier researhes are different from digital toologial aroahes. Thus the resent develos a new tye of loally finite toologial sae (or sae set toology, for short, SST) (see Definition.), whih is simler and more effiient than the axiomati toologial struture in [7], on a seially subdivided AC (for short, SAC) omlexes (see Definition.9). Indeed, an SST is different from an ALF sae in [7]: while an SST inludes a singleton, an ALF sae does not have the roerty. Besides, the tyes of oen sets of an SST are different from the ordinary ones in oint set toology. These oen sets are resented with a seial kind of neighborhoods onsisting of elements endowed with an SAC omlex instead of oints in a Hausdorff toologial sae. Conretely, in this state we need to oint out some differene between elements in the toology SST and oints in the general toology inluding Hausdorff toology. Further, omared with Mathematis Subet Classifiation. Primary 5F5; seondary 5A,5B5,5B5C Keywords. loally finite sae, Alexandroff sae, semi-t -searation axiom, abstrat ell omlex Reeived: July ; Aeted: 8 Aril Communiated by Dragan S. Dordević The author was suorted by Basi Siene Researh Program through the National Researh Foundation of Korea(NRF) funded by the Ministry of Eduation, Siene and Tehnology(RDAA98). address: sehan@bnu.a.kr (Sang-Eon Han)

2 S.-E. Han / Filomat :9 (), Khalimsky toologial saes and Marus-Wyse toologial saes, an SST an be substantially used for studying both ontinuous and digital saes so that it an be used in ontinuous and digital geometry suh as digital toology, omutational toology, omuter and disrete geometry, et. The aer [](see []) introdued the notion of T -searation axiom and the aer [9] established the notions of a semi-oen set and a semi-losed set. The axiom artially ontributed to the study of digital saes suh as one dimensional Khalimsky toologial sae. Motivated by these notions, the aer [] develoed the notion of semi-t -searation axiom. Further, this axiom has been often used in ure and alied toology inluding fuzzy toology. Moreover, the aer [5] makes the axiom more simlified (Proosition.). The aim of the aer is to study toologial roerties of an SST related to semi-t -axiom [], the axioms T and T and we rove that for a toologial sae (X, T) with X = the axioms T, semi-t and T are equivalent to eah other and if X, then we rove that semi-t does not imly T (see Theorems.7 and.9). The rest of this aer roeeds as follows. Setion realls some roerties of an SAC omlex whih is a seial kind of subdivision of an abstrat ell omlex. It also exlains a sae set toologial struture on an SAC omlex. Setion studies a relation between an SST and Khalimsky toologial saes. Setion roves that an SST satisfies the axioms T and further, shows that a onneted SST (X, T) with X satisfies the axiom semi-t, it does not satisfy the axiom T. Sae Set Toology on an SAC Comlex and its Proerty. Setion 5 onludes the aer with a summary. The works [ 8] develoed the notion of an AC omlex and studied its various roerties. Sine an AC omlex and ALF-sae [7] an be used for studying both ontinuous and digital saes, it an lay an imortant role in both lassial and digital geometry []. Thus we need to study its various roerties. Let R, Z and N be the sets of real numbers, integers and natural numbers, resetively. For a, b Z we use the notation [a, b] Z := {x Z a x b}. For a set X we follow the notation X as ardinality of the set. Let us onsider a neighborhood sae as a air S = (E, U) in the lassial textbook by Seifert and Threlfall [], where E is a nonemty set and U is a system of subsets of E, with the roerty that eah element e of E is ontained in some element of U, and that eah suh set belonging to U and ontaining e is alled a neighborhood of e. Based on the original version of an AC omlex in [] (see also [7]), we redefine below the notion of an abstrat ell omlex on the basis of a neighborhood relation instead of the bounding relation of [7]. Definition.. [] (see also []) An abstrat ell (for short, AC) omlex C = (E, N, dim) is a nonemty set E of elements rovided with () a reflexive, antisymmetri and transitive binary relation N E E alled the neighborhood relation, and () a dimension funtion dim: E I from E into the set I of non-negative integers suh that if a is an element of a neighborhood of b, then dim(a) dim(b). In Definition., the elements i of E = {i i M, M } are alled ells and the suersrit i of the ell i means its dimension, the subsrit of the ell means the only index for disriminating the i-dimensional ells, and the index sets M and M deend on the situation. In view of Definition., while the index set M i is finite, M need not be finite. i Remark.. () As for the terminology abstrat element of Definition., note that a ell of an abstrat ell omlex, unlike a Eulidean ell or a simlex, is never a subset of another ell. For instane, in Figure (a) let us onsider the obet as an AC omlex. Then it onsists of a two-dimensional ell (-ell for short or oen fae), five one-dimensional ells (-ells for short or oen line segments) and five -ells (or oints). Thus we observe that eah ell of an abstrat ell omlex is never a subset of another ell, whih imlies that an AC omlex is different from a simliial omlex.

3 S.-E. Han / Filomat :9 (), () In this state the neighborhood of Definition. does not require a toologial struture. () Comared with the dimension of a lassial toologial sae, the dimension of Definition. has its own roerty (see Definition.). Let us define the notion of smallest neighborhood of an element of an AC omlex. Definition.. [7](see also []) Let C = (E, N, dim) be an AC omlex. For an element a E let N(a, E) := N(a) = {b E (b, a) N}. Further, we denote by SN(a, E) := SN(a) the smallest N(a), alled a smallest neighborhood of a in E. In this state we need to oint out that the smallest neighborhood of Definition. is used without toology. Instead, it is only derived from an AC omlex. Let us now define the terminology adaent (or oins) between two ells of an AC omlex, as follows: Definition.. [7](see also []) Let C = (E, N, dim) be an AC omlex. For two distint elements a and b in E we say that a is adaent to (or oins) b if a SN(b) or b SN(a). For instane, onsider the obet in Figure (b) whih is an AC omlex. Then we say that eah of the -ells and is adaent to beause the faes and an be elements of SN( ) = {,, }. For eah ell of an AC omlex we an define its boundary as follows: Definition.5. [7](see also []) Let C = (X, N, dim) be an AC omlex, where X := { i i M, M }. For eah i ell i X we obtain its smallest neighborhood as follows: SN( i ) = {i } {i i is adaent to i, i > i}. In this state the notion of boundary of Definition.5 is not a toologial boundary. In view of Definition.5, we obtain the following: Remark.. For a -ell (or a oint) we say that =. In Figure (a) we an observe that onsists of five -ells and five -ones surrounding the -ell. The notion of a subdivision of a given omlex [] was often used in geometri toology, as in the ase of a triangulation, by means of a series of elementary subdivisions of its ells. Kovalevsky [8] established the notion of subdivision of an AC omlex. More reisely, for an m-ell m of an nd AC omlex with m n the aer [8] roeeded to subdivide the m-ell m. Using Remark. and Definition.5, we an now generalize the original version of a subdivision of m in [8] into that of m with m n whih is essentially used for establishing an SST (see Definition.9). Let us now suggest the orresondingly modified notion of elementary subdivision, as follows: Definition.7. [8] Let m be an m-ell of an n dimensional AC omlex C = (X, N, dim), where X := { i i M, M i } and m n. An elementary subdivision of the ell m relaes the ell m by two m-ells m, m and one (m )-ell m whose smallest neighborhood SN( m ) ontains both the m-ells m and m, while the elements m, m and m satisfy the onditions: ) ({ m, m, m }) = m ; ) m m ; ) m m. Examle.8. Figure (a) and (b) show a roess of an elementary subdivision of the roer -ell. The emhasized oints and in Figure (b) omose the -shere lying in the boundary. The -shere is sanned by the -ell. The original ell (or an oen -ell) is relaed in Figure (b) by the omlex { } { } { } whose boundary is the same as that of.

4 S.-E. Han / Filomat :9 (), (a) (b) AC omlex () r q X SAC omlex X (d) r q () () (e) (f) (g) Figure : (a)-(b): Configuration of a subdivision of the -ell followed from the subdivision of Definition.7; ()-(d) and (f)-(g): Proesses on onstruting SAC omlexes in terms of subdivisions of Definition.9. In order to roeed a seial kind of subdivision of an AC omlex, we need to reall tilings of R n, n N, as follows: The real lane an be a highly symmetri tiling made u of ongruent regular olygons. Only three kinds of regular tilings exist: those made u of equilateral triangles, squares or hexagons. An edge-to-edge tiling of a sublane in R is even less regular [7]. The only requirement is that adaent tilings only share full sides. Similarly, we an onsider a fae to fae quasirystallization of the D real sae [] and further, their analogy to R n, n. Motivated by these tilings and a (baryentri) subdivision of simliial omlexes [], we an establish the following: Definition.9. Let C = (X, N, dim) be an AC omlex, where X := { i i M, M }. Aording to Definition.7, roeed subdivisions of some m-ells in X to obtain an AC omlex (X, N, dim) suh that there is no i-ell i in X satisfying that i is artially mathed with i, where in M, where X := { i t i M, t M } with M M and M M. Then we all the subdivided AC omlex an SAC omlex. In view of Definitions. and.9, we observe that an SAC omlex is a finer AC omlex than given an AC omlex. Further, we need to oint out the inlusion M M of Definition.9 as follows: onsider a -ell without boundary. Then we an roeed a subdivision of by adding a -ell inside of. This roessing exlains the inlusion above. Examle.. Consider the AC omlex X of Figure (). Then, aording to Definition.9, we obtain an SAC omlex X derived from X. More reisely, while the obet X is an AC omlex, it annot be an SAC omlex beause some boundary of the -ells and are artially mathed with (see the -ells reresented by the line segments q and qr). Similarly, aording to Definition.9, Figure (f) and (g) show seial kinds of subdivisions of of Figure (e). Based on the smallest neighborhood of Definition., for an SAC omlex X we obtain a smallest neighborhood of an element i X as follows: Proosition.. Let C = (X, N, dim) be an SAC omlex, where X := { i i M, M }. For eah ell i X we obtain its smallest neighborhood on X as follows: SN( i ) = {i } {i i is adaent to (or oins) i, i > i}. The neighborhood relation SN of Proosition. fulfils Definition. suh as it is reflexive, transitive and antisymmetri. Hereafter, we will use the smallest neighborhood suggested in Proosition.. Furthermore, this neighborhood will be used for establishing a toology on an SAC omlex.

5 S.-E. Han / Filomat :9 (), Examle.. In Figure, aording to the dimensions,, and, we observe the orresonding smallest neighborhoods of given ells. In Figure () and () we obtain SN( ) = {, i, i i [, ] Z}, SN( ) = {,, } (see Figure ()). In artiular, assume that the otahedron without boundary in Figure () is formulated by the six oints i, i [, ] Z. Then it is exatly SN( ), where is generated by the four oints i, i [, ] Z. In Figure () onsider the roer tetrahedron generated by the five oints i, i [, 5] Z. Then SN( ) is the whole obet itself of Figure (). Dimension Grahial reresentation Smallest neighborhood () () 5 or () () 5 or 5 Figure : ()-() Various tyes of smallest neighborhoods whih an be onsidered in R and R. In artiular, the otahedron of Figure () is SN( ) [7], where the element is a retangle generated by the four oints,, and. Further, the tetrahedron of Figure () is the smallest oen set of the given -ell. Using the smallest neighborhood mentioned in Proosition., we now formulate a toology on an SAC omlex X, named by a sae set toology on X, as follows: Definition.. Let C = (X, N, dim) be an SAC omlex. Let S := (X, U) be a binary set, where U is the set of all SN(x), x X. Then we obtain the toology on X indued by the set U as a base, denoted by (X, T). Further, we all this toology T a sae set toology on X, briefly SST on X. For an SAC omlex X and an element x X the neighborhood SN(x) suggested in Proosition. is exatly a smallest oen set of an element x in the SST, (X, T). Remark.. () Based on Proosition. and Definition., we now onsider a toologial neighborhood of an element of an SST. Further, for an SST, (X, T), the smallest neighborhood of Proosition. is exatly that of Definition.5 and further, it is the smallest oen set of an element of (X, T). Furthermore, if X, then a onneted SST (X, T) annot be a disrete toologial sae. () For an SST (X, T) onsider two distint elements x and y whih are not adaent to eah other. Then there are smallest oen neighborhoods of the elements, denoted by SN(x) and SN(y) in T, suh that y SN(x) and x SN(y).

6 . A Relation Between SSTs and Khalimsky Saes S.-E. Han / Filomat :9 (), For an SAC omlex C := (X, N, dim) in order to define a dimension of an element X, we use the following notation. Consider a sequene anbn Nk of airwise distint ells of X in whih eah ell belongs to the smallest neighborhood of the next one: a SN(b) and a b; b SN() and b et. We shall all it the neighborhood ath from a to k. The number of ells in the sequene minus one is alled the length of the neighborhood ath. We an now define the notion of dimension of a ell C by using the notion of the neighborhood ath. Definition.. Let C := (X, N, dim) be an SAC omlex. For X the dimension dim(, C) of the ell of C is the length of a longest neighborhood ath from to any element of C. For a ell that does not belong to any smallest neighborhood of another ell the length and the dimension are zero. The dimension of an SAC omlex is the greatest dimension of its ells. Sine the neighborhood relation is transitive, smallest neighborhoods omosing a neighborhood ath are interloked in eah other in the way symbolially shown in Figure. Hene, the dimension of a ell is its deth in the rater of interloked neighborhoods. Figure : Symboli reresentation of interloked smallest neighborhoods of the ells vn f NN [7, 8]. This definition is equivalent to the known notion of the dimension or height of an element of a artially ordered set (oset) as defined in the theory of oset toology [] and of A-saes []. Figure shows some examles of the smallest neighborhoods of ells of different dimensions while using the grahial reresentation suggested in [7]. Note that Figure is only an examle. Cells may have quite different numbers of other ells in their smallest neighborhoods. The grahial reresentations an also vary orresondingly as it was seen in Figure () and (): for instane, ells of dimension an be reresented as olygons with various numbers of elements (see Figure ()). To erform toologial and geometrial alulations with abstrat ell omlexes, it is neessary to assign names to the ells. One of the ossibilities of doing this onsists in introduing oordinates. Remark.. Combinatorial oordinates were alled toologial oordinates in the earlier ubliations of [8]. The reent term ombinatorial oordinates was introdued in [7]. Consider an AC omlex C = (E, N, dim) whose elements omose a sequene E = (e, e, e,, e m ), m. The smallest neighborhoods SN(e i ) are the following: SN(e i ) = {e i, e i, e i+ } E if i is even and SN(e i ) = {e i } if i is odd. The dimension of e i is defined aording to Definition.: dim(e i ) = if i is even and dim(e i ) =

7 S.-E. Han / Filomat :9 (), if i is odd. Thus C beomes a one-dimensional AC omlex onsisting of a single adaent ath (Definition.). We shall all suh a omlex a ath omlex. We rovide a ath omlex C with a oordinate funtion X : E Z assigning subsequent integer numbers (not neessarily ositive) to subsequent ells of E in suh a way that a ell of dimension (a -ell for short) obtains an even number while a ell of dimension (-ell) obtains an odd one. We all the numbers ombinatorial oordinates in C. Definition.. [7] A ombinatorial oordinate axis A = (E, X) is a one-dimensional ath omlex rovided with a oordinate funtion X : E Z assigning subsequent integer numbers to subsequent ells of E. It is also alled a oordinate axis for short. The set of smallest neighborhoods of a ath omlex an be obviously a base for an SST. Thus a oordinate axis is an SST. It is easily seen that the Khalimsky line [] being regarded as a toologial sae is equivalent to the oordinate axis sine both have the same base. The differene between a oordinate axis and the Khalimsky line is the resene of dimensions of ells of the axis. Another less imortant differene is that elements of the Khalimsky line being not regarded as a quotient sae of a Eulidean line are integer numbers, while elements of a oordinate axis are ells with integer numbers assigned to them. An SST of dimension n > an be defined as the Cartesian rodut of n oordinate axes. It ossesses the rodut toology derived from the toology of a oordinate axis in terms of the smallest neighborhood suggested in Proosition., as follows: Definition.. An nd Cartesian SAC omlex C n = (E n, N, dim) is the Cartesian rodut of n ombinatorial oordinate axes A i = (E i, X i ); i =,,, n; rovided with a oordinate funtion X n. The set E n = E E E n. A ell of C n is an n-tule, i.e. an ordered sequene of ells a i of the axes: a i A i. For an n-tule = (a, a,, a n ) E n we define its smallest neighborhood as SN() = SN (a ) SN (a ) SN n (a n ) where SN i (a i ) is the smallest neighborhood of a i in A i. The oordinate funtion X n : E n Z n assigns the n-tule X n () = (X (a ), X (a ),, X (a n )) to a ell = (a, a,, a n ) of E n. The dimension funtion dim assigns to eah ell E n its dimension aording to Definition.. It is equal to the number of odd oordinates X i (a i ) of f (,7) (, ) (,5) (,) (a) (b) Figure : (a): Grahial reresentations of a ombinatorial oordinate axis; (b): a two-dimensional Cartesian AC omlex [8]. Figure (a) shows a ombinatorial oordinate axis. The -ells of the axis have even oordinates, while the -ells have odd oordinates. Figure (b) shows a two-dimensional AC (or SAC) omlex C whih is the Cartesian rodut of two ombinatorial oordinate axes. Two-dimensional ells (-ells for short) are reresented in Figure (b) as interiors of squares; one-dimensional ells (- ells) as line segments whih are sides of the squares and -ells as rossing oints of lines whih is the same as the orners of squares.

8 S.-E. Han / Filomat :9 (), Some ells are emhasized in Figure (b), while a square is shaded, a line is bold or a -ell is reresented as a small blak square. Emhasized ells are aomanied by their ombinatorial oordinates aording to the above Definition.. A two-dimensional ell of a two-dimensional SAC omlex is often alled a ixel, a -ell is alled a rak and a -ell a oint. Thus the ell with oordinates (, ) is a horizontal and that with oordinates (, 5) a vertial rak. In relation to Theorem.5 below, let us now reall the notion of Khalimsky line toology on Z denoted by (Z, κ) []. Motivated by the A-sae in [], Khalimsky line toology κ on Z is indued by the family of the subset {{n + }, {m, m, m + } m, n Z} whih indues the oen sets of κ. In addition, the notion of T -searation axiom of a toologial sae (X, T) was develoed by Levine [](see also []), whih means that eah singleton of (X, T) is either oen or losed. It turns out that (Z, κ) is a T -sae. In Khalimsky toology the set [a, b] Z = {n Z a n b} is onsidered as a subsae of (Z, κ), is alled a Khalimsky interval. Theorem.5. The Khalimsky line toologial sae is equivalent to an SST of Z. Proof: Sine the Khalimsky line (Z, κ) onsists of alternating oen and losed oints, every element e of Z has its smallest neighborhood SN(e) whih is {n, n, n + } if e = n, n Z or {e} if e = n, n Z. These smallest neighborhoods an be a base for an SST of Z (a) 5 5 (b) () Figure 5: Different grahial reresentations of one and the same sae: (a) A mixed, a losed and an oen oint in the reresentation usual for the Khalimsky D sae; (b) Smallest neighborhoods of the oints i, i {,, }, of (a) [8]; () The same oints i of (b), i {,, } and their smallest neighborhoods SN( i ) in the reresentation usual for SAC omlexes. Unlike the axioms T, T, T, the axiom T does not have the rodut roerty. More reisely, let (Z n, κ n ) be the rodut toologial sae of n-oies of (Z, κ), n. The rodut toologial sae (Z, κ ) is alled the digital lane. Then it does not satisfy the axiom T [5]. Consequently, if n, then the Khalimsky n-sae (Z n, κ n ) is a T sae rather than a T sae sine unlike the Khalimsky line toologial sae (Z, κ) there are singletons of Z n with n whih are neither oen nor losed [5]. Furthermore, using several kinds of ontinuities and homeomorhisms in [9], we an also study some subsaes of the Khalimsky nd sae []. Let us reall that a oint x = (x, x,, x n ) Z n of the Khalimsky nd sae (Z n, κ n ), is oen if all oordinates are odd, and losed if all oordinates are even []. These oints are alled ure and other oints in Z n are alled mixed. For a subset X of Z n let us onsider a subsae (X, κ n ) of the nd Khalimsky X sae (Z n, κ n ). Figure 5 shows two different grahial reresentations of one and the same sae. Figure 5(a) shows a two-dimensional Khalimsky sae with three emhasized oints: = (, ) is a losed oint, = (5, 7) is an oen and = (, ) a mixed one. Figure 5(b) shows the smallest neighborhoods of these three oints.

9 S.-E. Han / Filomat :9 (), Figure 5() shows the same sae reresented as a two-dimensional Cartesian omlex in a grahial reresentation usual for an SST. The losed oint = (, ) is here regarded as a -ell and is reresented as a small square. The oen oint = (5, 7) is regarded as a -ell and is reresented as the interior of a square. This means that the square ontains neither its sides nor its orners. The mixed oint = (, ) an be onsidered a -ell or a horizontal rak and is reresented here as a horizontal line segment. The smallest neighborhoods of these ells are easier to be deteted than in the reresentation of Figure 5(b) beause grahial reresentations of two inident ells touh eah other. The smallest neighborhood looks in this reresentation as a onneted subset of the lane, whih is not the ase in the reresentation of Figure 5(b). Lemma.. Sine Definition. uses only the notion of smallest neighborhoods, it an be alied to any SST for defining the dimensions of its elements and the dimension of the sae. Theorem.7. Every onneted SST ontaining at least two elements and satisfying the searation axiom T one-dimensional sae. Proof: An SST, (X, T), with the searation axiom T has er definition only two kinds of elements: oen or losed ones []. If the sae is onneted, then the smallest neighborhood of a losed element ontains at least one oen element. Let X be a losed element of X and o X one of the oen elements in SN(). Sine the neighborhood relation is a reflexive artial order and o, we an write < o. Then there is no element a X suh that < a < o beause in this ase a an be neither oen nor losed, thus it would be alled a mixed element. But there are no mixed elements in a T sae. Therefore, the longest neighborhood ath has the length and eah oen element has dimension. The dimension of an SST is aording to Lemma. the greatest dimension of its elements. Therefore the dimension of (X, T) is equal to. is a. Toologial Proerties of an SST This setion investigates toologial roerties of an SST and roves that a onneted SST is a T -sae and further, roves that a onneted SST is an good examle showing that the axiom semi-t does not imly T. Sine the notions of semi-oen and semi-losed an be often used in ure and fuzzy toology, we need to reall them as follows: Definition.. [9] Let (X, T) be a toologial sae. A subset A of X is alled semi-oen if there is an oen set O T suh that O A Cl(O), where Cl(O) means the losure of the set O. A subset F X is alled a semi-losed set of a toologial sae (X, T) if X \ F is semi-oen in (X, T). The notion of semi-oen of the subset A in Definition. an be equivalently reresented as follows: A Cl(Int(A)) [9]. Further, the notion of semi-losed of the subset A in Definition. an be equivalently reresented as follows: there exists a losed set F in T suh that Int(F) A F or Int(Cl(A)) A [9]. Lemma.. Let (X, T) be a onneted SST with X, where X := { i i M, M }. Let X t be a totally ordered subset of the given artially ordered set X. For an i-ell i in Xt we obtain its losure in (X, T) as follows: Cl( i ) = { i }; if i is a minimal element of Xt u to a dimension, and { i, t t is adaent to i, t i} : else Proof: If i is a minimal element of X t u to a dimension, then it is losed in (X, T) beause the set X \ { i } is a union of smallest oen neighborhoods of the other elements i in X with i i, whih imlies that Cl( i ) = {i }.

10 S.-E. Han / Filomat :9 (), If the given i-ell i is not a minimal element of Xt, then Cl( i ) should be the set {i, t t is adaent to i, t i} beause it is the largest subset of X ontaining the element i suh that X \ Cl(i ) should be oen in (X, T). For instane, onsider a -ell d 8 and a -ell in X of Figure. Then we an observe that Cl(d 8 ) = {d 8,, } and Cl( ) = {, d, d, d 7,, }. Corollary.. Let (X, T) be a onneted SST with X, where X := { i i M, M }. Let X t be a totally ordered subset of the given artially ordered set X. Let i be an i-ell in X whih is not a maximal element of Xt u to dimension. Then Cl(X \ Cl({ i }) = X. For instane, onsider the element d in X of Figure. In the SST (X, T) we obtain that Cl({d }) = {d, } and Cl(X \ Cl({d })) = X (observe the elements or beause Cl({ }) inludes the set Cl({d }). It is well known that the Khalimsky line toology [](see also [5]) and the Marus-Wyse toologial sae [5] satisfies the searation axiom T. As a weaker form of the axiom T the following notion was develoed in []. Let us now investigate its roerties. Hereafter, we denote by scl(a) the intersetion of all semi-losed sets ontaining A, i.e. scl(a) means the semi-losure of A. Definition.. [] Let (X, T) be a toologial sae. A subset A of X is alled semi-generalized losed if scl(a) O holds whenever A O and O is semi-oen in (X, T). Definition.5. [](see also [, ]) We say that a toologial sae (X, T) satisfies the semi-t (or a semi-t sae) if every semi-generalized losed set in (X, T) is semi-losed. The aer [5] haraterizes the axiom semi-t as follows. Proosition.. [5] We say that a toologial sae (X, T) satisfies the semi-t sae) if and only if every singleton of X is either semi-oen or semi-losed. searation axiom searation axiom (or a semi-t In view of Definition., it is lear that an oen set (res. a losed set) imlies a semi-oen set (res. a semi-losed set). Thus we obtain that the axiom T imlies the axiom semi-t. In addition, a toologial sae (X, T) is alled loally finite (for short, LF) if eah element of X has a finite neighborhood and it is alled an Alexandroff toologial sae (A-sae for short) if eah oint (or element) of X has its smallest oen neighborhood in X []. For a onneted SST, (X, T), if {i i X}, then (X, T) annot satisfy the T -searation axiom. However, aording to Proosition., an SST has the following roerty. Theorem.7. A onneted SST satisfies the searation axiom T. Proof: For an SST, (X, T), take two distint elements in X suh as t and t with t t. By Proosition., we obtain SN( t ) or SN( t ) suh that at least either of them does not ontain the other oint t or t. More reisely, aording to Proosition., we an onsider two ases as follows: (Case ) Assume that t is adaent to t. Then it is obvious that t t. By Proosition., we obviously obtain SN( t ) and SN( t ) suh that t SN( t ) or t SN( t ). More reisely, for onveniene assume that t t. Then, by Proosition., it is obvious that SN( t ) does not ontain the element t. (Case ) Assume that t is not adaent to t. Then, by Proosition., we obtain SN( t ) and SN( t ) suh that t SN( t ) and t SN( t ), whih omletes the roof. Examle.8. Consider an SST (X, τ ) (res. (Y, τ )) endowed with the SAC omlex X(res. Y) of Figure, where X omoses six -ells suh as { i i [, ] Z }, twelve -ells suh as {d i i [, ] Z } and six -ells suh

11 S.-E. Han / Filomat :9 (), as { i i [, ] Z }, and Y omoses six -ells and six -ells without -ell. Then we observe that both (X, τ ) and (Y, τ ) are onneted. Take two distint elements x and y in X or Y. For onveniene, onsider the two elements in X. Aording to the adaeny of the two elements, we an onsider the following two ases as follows: (Case ) Assume that x is adaent to y in X. In this ase it is obvious that dim x annot be equal to dim y. Thus, for onveniene we may assume that dim y dim x. Then we obtain SN(y) suh that x SN(y), whih imlies that (X, τ ) satisfies the axiom T. For instane, take two elements d 8 := y and := x in X of Figure. This hoie suorts that x SN(y), whih imlies that (X, τ ) satisfies the axiom T beause SN(d 8 ) = {d 8, } and SN( ) = {, d, d 7, d 8,, } in (X, τ ). As another examle, take two elements := y and d := x in Y of Figure. This hoie also exlains that x SN(y), whih imlies that (Y, τ ) satisfies the axiom T beause SN(d ) = {d,, }. (Case ) Assume that x is not adaent to y in X. In this ase, by Remark.(), regardless of the dimensions of the two elements x and y we an take two oen sets SN(x) and SN(x) in (X, τ ) suh that y SN(x) and x SN(y). For instane, take two elements := y and d 8 := x in X of Figure. This hoie suorts that y SN(x) and x SN(y) beause SN( ) = { } SN(d 8 ) = {d 8, }. Similarly, onsider two elements d := y and d := x in Y of Figure. This hoie shows that y SN(x) and x SN(y) beause SN(d ) = {, d, } SN(d ) = {, d, } in (Y, τ ). d g d g d g d d 8 9 d 7 d g d d d g d g5 5 d 5 X Y g d g d g d g 5 d 5 d g d g Figure : Exlanation of the axioms semi-t and T of SST. As mentioned above, it is obvious that the T -searation axiom imlies the semi-t However, motivated by Theorem.7, we now need to omare the axiom semi-t Theorem.9. () For a toologial sae (X, T) with X = the axioms T, semi-t other. () Let (X, T) be a toologial sae with X. Then the axiom semi-t searation axiom. with T, as follows: and T are equivalent to eah does not imly T. Proof: Let (X, T) be a toologial sae suh that X =. First, let us rove that the axioms T and semi-t are equivalent to eah other. Assume that (X, T) satisfies the axiom T and X := {a, b}. Then the singleton omosed one of the two distint elements should be oen or losed. More reisely, the axiom T imlies that a Cl({b}) or b Cl({a}). In ase a Cl({b}), the singleton {b} is losed beause X =, whih means that {a} is oen. Similarly, in ase b Cl({a}), the singleton {a} is losed, whih means that {b} is oen, whih imlies that (X, T) satisfies the axioms semi-t and T. Conversely, assume that (X, T) satisfies the axiom semi-t and {a} is semi-oen. Hene there is an oen set O T suh that O {a} Cl(O), the oen set O should be the singleton {a}, whih imlies that (X, T) satisfies the axiom T. Seond, assume that {a} is semi-losed. Sine there is an oen set U T suh that U X \ {a} := {b} Cl(U), the oen set U should be the singleton {b}, whih imlies that (X, T) satisfies the axiom T. Let us now rove that T and T hyothesis it is obvious that T and T are equivalent to eah other. Using the same method as above, with the are also equivalent to eah other. does not imly T, we onsider a ounter examle of the () In order to rove that the axiom semi-t assertion. Let (X, T) be a onneted SST with X, where X := { i i M, M }. Let X t be a totally

12 S.-E. Han / Filomat :9 (), ordered subset of the given artially ordered set X. Let i (res. i ) be a minimal (res. maximal) element of X t u to a dimension. Then the ells i in X have the following roerty. (Case ) The singleton { i } is semi-losed. Let us rove this assertion. Sine the given sae (X, T) is onneted and X, we may onsider two ases, as follows: (Case ) In ase {i i X} =, aording to Proosition., the assertion is trivial beause the singleton { i } is losed and {i } is oen. Consequently, {i } (res. {i }) is semi-losed (res. semi-oen). (Case ) For eah i with i i i the singleton { i } is semi-losed. Let us rove this assertion. Assume that {i i X}. () By Proosition., we observe that the singleton { i } is losed beause the set X \ {i } is a union of smallest oen neighborhoods of the other elements i in X with i i, whih imlies that X \ {i } is an oen set. Consequently, the singleton { i } is losed, so semi-losed. () Consider the singleton { i } with i i i. Sine the singleton { i } is neither oen nor losed, let us now take an oen set X \ Cl({ i }) := O in (X, T). Then we obtain that X \ Cl({ i }) X \ {i } Cl(X \ Cl({i })), beause, aording to Corollary., Cl(X \ Cl({ i })) is the whole set X, whih omletes the roof. Let us now onsider an examle guaranteeing. Consider the SST (X, T) endowed with the SAC omlex X in Figure. Then we observe that the element d 7 has dimension and further, X \ {d 7,, } X \ {d 7 } Cl(X \ {d 7,, }) = X, whih imlies that the singleton {d 7 } is semi-losed beause Cl(d 7 ) = {d 7,, }. (Case ) The singleton { i } is semi-oen. Let us rove this assertion. Aording to Proosition., the singleton { i } is obviously oen, whih imlies that it is obviously semi-oen. 5. Conlusion We have investigated various roerties of an SST whih an be suessfully emloyed for reresenting loally finite toologial saes in a omuter and for solving toologial roblems. Indeed, the theorem an lay an imortant role in lassial, omuter, disrete and digital geometry as well as digital toology beause it has its own roerty different from the lassial one. As mentioned in Setions and, it turns out that SST is an Alexandroff toologial struture with the axioms T and semi-t [], whih an ontribute to alied toology relevant to omuter siene. Finally, an SST an be used for studying both ontinuous and digital saes so that it lays an imortant role in both lassial and digital toology and further, omuter and disrete geometry []. Referenes [] P. Alexandorff, Diskrete Räume, Mat. Sb. (97) [] P. Bhattaharyya, B.K. Lahiri, Semi-generalized losed set in toology, Indian J. Math. 9 (987) [] G. Birkhoff, Lattie Theory, Amerian Mathematial Soiety, 9. [] V.A. Chatyrko, S.E. Han, Y, Hattori, Some remarks onerning semi-t saes Filomat 8() () -5. [5] M.C. Cueva, R.K. Saraf, A researh on haraterizations of semi-t saes, Divulgaiones Mathemǎtias 8() () -5. [] W. Dunham, T -saes, Kyungook Math. J. 7 (977), -9. [7] Grunbaum, Branko and G.C.Sheard, Tilings and Patterns, New York, W. H. Freeman & Co., 987. [8] N. Hamllet, A orretion to the aer Semi-oen sets and semi-ontinuity in toologial saes by Norman Levine, Proeeding of Amer. Math. So. 9() (975) 58-. [9] S.-E. Han, KD-(k, k )-homotoy equivalene and its aliations Journal of Korean Mathematial Soiety, 7(5) () -5. [] S.-E. Han, Continuity of mas between axiomati loally finite saes and its aliations, Intenational Journal of Comuter Mathematis 88() () [] S.-E. Han, Study of toologial saes with the semi-t searation axiom, Honam Math. Jour. 5 () () 77-7.

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