finite topological spaces Julian L. Cuevas Rozo1,∗ , Humberto Sarria Zapata1
1 Cienc. Ex. Fis. Nat. 41(158):127-136, enero-marzo de 2017 Rev. Acad. Colomb. MatricesBogot´ associated to finite topological spaces Dpto. de Matem´ aticas, Facultad de Ciencias, Universidad Nacional de Colombia, a, Colombia doi: http://dx.doi.org/10.18257/raccefyn.409
Abstract Original article
Mathematics
The submodular functions have shown their importance functions in the study and characterization of Computation of matrices and submodular values associated to multiple properties of finite finite topological spaces, from numeric values provided by such functions topological spaces Computation of matrices and functions (Sarria, Roa & Varela, 2014). However, the submodular calculation of these values has been values performed Computation of matrices and submodular functions values associated to manually or even using Hasse diagrams, which we present some 1,∗ is not practical. In this article, 1 Julian L. finite Cuevas Rozo , Humberto Sarria spaces Zapata associated to finite topological topological spaces 1 algorithms that let us calculate some kind of polymatroid functions, specifically fU , fD , f and rA , Dpto. de Matem´ aticas, Facultad de Ciencias, Universidad Nacional de Colombia, Bogot´ a, Colombia
which define a topology, by using topogenous1,∗matrices. Julian L. Cuevas Rozo , Humberto Sarria Zapata1 1
Dpto. de Matem´ aticas, Facultad de Ciencias, Universidad Nacional de Colombia, Bogot´ a, Colombia Key words: Finite topological spaces, submodular functions, topogenous matrix, Stong matrix. Abstract
nd submodular functions values associated to C´ alculo de matrices y funciones submodulares asociadas espacios ogicos finitos The submodular functions have shown their importance in thea study and topol´ characterization of Abstract nd submodularspaces functions values associated to from numeric values provided by such functions te topological multiple properties of finite topological spaces, te topological spaces Resumen (Sarria, Roa &1functions Varela, 2014). However, calculationinofthe these values been performed The submodular have shown theirthe importance study andhas characterization of 1,∗
vas Rozo , Humbertomanually Sarria Zapata or even using Hasse diagrams, spaces, which is not numeric practical.values In this article,by wesuch present some multiple properties of finite topological from provided functions 1 Bogot´ e Ciencias, Universidad Nacional de Colombia, a, Colombia vas Rozo1,∗ , Humberto Sarria Zapata Las funciones submodulares han mostrado su importancia en el specifically estudio y flaU , caracterizaci´ algorithms that& letVarela, us calculate some kind of polymatroid functions, fD , performed f and rAo,n (Sarria, Roa 2014). However, the calculation of these values has been e Ciencias, Universidad Nacional de Colombia, Bogot´ aby , Colombia m´ ultiples propiedades de los espacios topol´omatrices. gicos finitos, a partir de valores num´ericos proporwhich define a topology, using topogenous manually or even using Hasse diagrams, which is not practical. In this article, we present some cionados por dichas funciones (Sarria, Roa & Varela, 2014). Sin embargo, el c´alculo de ´estos algorithms that let us calculate some kind of polymatroid functions, specifically fU , fD , f and rA , valores se ha Finite realizado manualmente e submodular incluso haciendo uso detopogenous diagramasmatrix, de Hasse, lo que no es Key topological functions, Stong matrix. whichwords: define a topology, by usingspaces, topogenous matrices. pr´actico. En este art´ıculo, presentamos algunos algoritmos que nos permiten calcular cierta clase hown their importance in the study and characterization of dealculo funciones polimatroides, espec´ıficamente fU , fD , fasociadas y rA , las a cuales definen unaogicos topolog´ ıa, por C´ de matrices y funciones submodulares espacios topol´ finitos Key words: Finitecharacterization topological spaces, gical spaces, from numeric values provided by such functions hown their importance in the study of submodular functions, topogenous matrix, Stong matrix. medio del usoand de matrices topog´eneas. However, the from calculation these provided values hasbybeen gical spaces, numericofvalues suchperformed functions Resumen C´ a lculo de matrices y funciones ogicos finitos rams, which is not practical. In this article, we present somesubmodulares asociadas a espacios topol´ However, the calculationPalabras of these values has been performed clave: Espacios topol´ ogicos finitos, funciones submodulares, matriz topog´enea, matriz e kind of polymatroid functions, specifically f , , f , f and r rams, which is not practical. In this article, Uwe present some A D de Stong. Las funciones submodulares han mostrado su importancia en el estudio y la caracterizaci´on Resumen opogenous matrices. functions, me kind of polymatroid specifically fU , fD , f and rA , m´ ultiples propiedades de los espacios topol´ogicos finitos, a partir de valores num´ericos proporopogenous matrices. cionados por dichas funciones (Sarria, Roa su & Varela, 2014). embargo, alculo de ´estos Lastopogenous funciones submodulares han mostrado importancia en Sin el estudio y el la c´ caracterizaci´ on ces, submodular functions, matrix, Stong matrix. valores se ha realizado manualmente e incluso haciendo uso de diagramas de Hasse, lo que no es m´ ultiples propiedades los matrix. espacios topol´ogicos finitos, a partir de valores num´ericos proporaces, submodular functions, topogenous matrix, de Stong pr´ actico. En este art´ ıculo, presentamos que nos cierta cionados por dichas funciones (Sarria, algunos Roa & algoritmos Varela, 2014). Sinpermiten embargo,calcular el c´alculo de ´eclase stos s submodulares asociadas a espacios topol´ ogicos finitos de funciones polimatroides, espec´ıficamente fU ,haciendo fD , f y ruso cuales definen una topolog´ ıa,nopor A , las valores se ha realizado manualmente e incluso de diagramas de Hasse, lo que es provide s submodulares asociadas a espacios topol´ o gicos finitos Introduction problem still unsolved. Moreover, such matrices medio del En usoeste de matrices eneas. algunos algoritmos pr´ actico. art´ıculo, topog´ presentamos que nos permiten calcular cierta clase all the information about the topology of a finite space de funciones polimatroides, espec´ıficamente fU , fD(Shiraki, , f y rA , las cuales definen topolog´ por objects in 1968), showing theuna relevance ofıa, these Palabras clave: Espacios topol´ o gicos finitos, funciones submodulares, matriz topog´ e nea, matriz Alexandroff proved that finite topological spaces are medio del uso de matrices topog´ e neas. mostrado su importancia en el estudio y la caracterizaci´on the topological context. Stong. valores in correspondence one-to-one with proporfinite ios topol´ogicos finitos, adepartir ericos mostrado su importancia en el de estudio y num´ la caracterizaci´ on preorders (Alexandroff, 1937), showing such spacesfinitos, can be clave: Espacios topol´ gicos funciones submodulares, matriz topog´enea, matriz ria, Roaogicos & Varela, Sin el c´ athat de ´oestos ios topol´ finitos,2014). aPalabras partir deembargo, valores num´ elculo ricos proporRecently, connections between submodular functions and viewed from other mathematical structures. Likewise, Shide Stong. eria, e incluso haciendo uso de diagramas de Hasse, lo que no es Roa & Varela, 2014). Sin embargo, el c´ alculo de ´estos finite topological spaces have been developed (Abril, raki named topogenous objects mos algunoshaciendo algoritmos permiten calcular cierta clase te e incluso usoque deasnos diagramas dematrices Hasse, lothe que no es worked by 2015), (Sarria et al., 2014), allowing to interpret many Krishnamurthy, inpermiten andefinen attempt to topolog´ count the topologies that camente fU , falgoritmos las nos cuales una ıa,clase por mos algunos calcular cierta D , f y rA ,que topological concepts through numeric values provided by can defined on a finite setuna (Krishnamurthy, as. camente fU , fDIntroduction , f yberA , las cuales definen topolog´ıa, por 1966), a problem such associated functions, which issuch reallymatrices important if we still unsolved. Moreover, provide eas. want to mechanize the verification of topological properall the information about the topology of a finite space * Correspondence: J. L. Cuevas Rozo,
[email protected], cos finitos, funciones submodulares, matriz topog´enea, matriz Received xxxxx XXXX; Accepted xxxxx XXXX. ties on subsets ofshowing an arbitrary finite space. (Shiraki, 1968), the relevance ofmatrices these objects in Introduction still unsolved. Moreover, such provide Alexandroff proved that finite topological icos finitos, funciones submodulares, matriz topog´ enea, matrizspaces are problem the topological context. In view of the above, we regard some matrices, as toin correspondence one-to-one with finite preorders all the information about the topology of a finite space pogenous and Stong matrices defined in sections 2 and 3, (Alexandroff, 1937),that showing such spaces can are be (Shiraki, 1968), showing the relevance of these objects in Alexandroff proved finitethat topological spaces which areconnections useful, for between example,submodular in the study of lattice Recently, functions andof viewed from other mathematical Likewise, Shi- the topological context. in correspondence one-to-one structures. with finite preorders all topologies on aspaces fixed have set, and we developed link such matrices finite topological been (Abril, raki named as topogenous matrices objects worked (Alexandroff, 1937), showing thatthesuch spaces can by be 2015), rA ), ffunctions with particular submodular (f (Sarria et al.,between 2014),functions allowing toU ,interpret many D , f and Recently, connections submodular and Krishnamurthy, in an attempt to count the topologies viewed from other mathematical structures. Likewise, that Shi- topological important in the finite topological spaces context, by alconcepts through numeric values provided by finite topological spaces have been developed (Abril, problem stillasunsolved. such matrices provide can be defined on a finiteMoreover, set (Krishnamurthy, 1966), raki named topogenous matrices the objects worked bya such gorithms created to improve the computations shown in associated functions, which is really important if we 2015), (Sarria et al., 2014), allowing to interpret many all the information the to topology of topologies a finiteprovide space problem still unsolved. Moreover, suchthe matrices Krishnamurthy, in anabout attempt count that want (Abril, 2015) and (Sarria et al., 2014). to mechanize the verification of topological proper* Correspondence: J. L. Cuevas Rozo,
[email protected], (Shiraki, 1968),onshowing objects ina topological concepts through numeric values provided by information aboutthe therelevance topologyofofthese a finite space canthe be defined a finite set (Krishnamurthy, 1966), spaces are all Received xxxxx XXXX; Accepted xxxxx XXXX. ties on subsets offunctions, an arbitrary finite space.important if we which is really the topological context. (Shiraki, 1968), showing the relevance of these objects in such Fromassociated now on, we shall use exclusively the symbol X to e spaces preorders are want to amechanize the verification of1 ,topological properthe topological context. * Correspondence: J. L. Cuevas Rozo,
[email protected], . . . , xn }, unless exdenote set of n elements X = {x can be eaces preorders Received xxxxx XXXX; Accepted XXXX. functions and ties on subsets of an arbitrary finite Ispace. Recently, connections betweenxxxxx submodular = {1, 2, . . . , n} and plicitly stated otherwise. We define n ikewise, aces canShibe *Corresponding autor: finite topological spaces have been developed (Abril, a permutation σ of In , we use Pσ to denote the matrix Julian L. Cuevas Rozo,
[email protected] tsikewise, workedShiby Recently, connections between submodular functions and for 2015), (Sarria et spaces al., 2014), to interpret(Abril, many whose finite topological haveallowing been developed entries satisfy Received: September 30, 2016 pologies that ts worked by topological concepts through numeric values provided by Accepted: March 6, 2017 2015), (Sarria et al., 2014), allowing to interpret many hy, 1966), a pologies that such associated functions, which is really important if we [Pσ ]ij = δ(i, σ(j)) topological concepts through numeric values provided by hy, 1966), a to mechanize the verification topological properExampl such associated functions, which is of really important if we
[email protected], want 127 where X of anthe arbitrary finiteofspace. wantontosubsets mechanize verification topological proper- where δ is the Kronecker delta.
[email protected], ties ties on subsets of an arbitrary finite space. T =
, as to2 and 3, attice of matrices and rA ), t, by alhown in
ol X to nless ex, n} and e matrix
1 0 0 0 1 0 which are useful, for example, in the study of lattice of 6 . . , xsuch exdenote a set of = {x n }, unless 0 1 0 •x 0 0 0 all topologies onnaelements fixed set,Xand we1 , .link matrices 2, . . . , n} and plicitly stated otherwise. Wefunctions define In(f= ,{1, •x2 0 0 1 0 0 0 with particular submodular U fD , f and rA ), T = for a permutation σ of Itopological the matrix X n , we use Pspaces σ to denote 0 1 0 1 0 1. important in the finite context, by al Fis. Nat. 41(158):127-136,enero-marzo de 2017 Cuevas Rozo JL, Sarria Zapata H Rev. Acad. Colomb. Cienc. Ex. whose entries satisfy 1 0 0 0 1•x05 gorithms created to improve the computations shown in doi: http://dx.doi.org/10.18257/raccefyn.409 •x 0 03 0 0•x 0 1 (Abril, 2015) and (Sarria et al., 2014). 1 [Pσ ]ij = δ(i, σ(j)) space (X, T ) the topological From now on, we shall use exclusively the symbol X to Example 0.4. Consider 1 0 0 0 1 0 where δa issetthe and denote of Kronecker n elementsdelta. X = {x1 , . . . , xn }, unless ex- where X = {a, b, c, d,e} 0 1 0 0 0 0 plicitly stated otherwise. We define In = {1, 2, . . . , n} and 0 0 1 0 0 0 T = {∅, {b} , {b, d} , {d, e} , {b, d,e} , {a, b, d}, TX, {d} = for a permutation σ of In , we use Pσ to denote the matrix 0 1 0 1 0 1. b, c, d} , X} b, d, e} , {a, whose entries satisfy matrix 1 0 {a, Topogenous 0 0 1 0 0 0 of0 the 0 elements, 0 1 For the following orderings we obtain the [Pσ ]ij = δ(i, σ(j)) Given a finite topological space (X, T ), denote by Uk the Example respective topogenous matrices as can be verified by T cal0.4. Consider the topological space (X, ) minimal set containing xk : culating the{a, minimal basis where δ isopen the Kronecker delta. where X= b, c, d, e} andin each case: (x1,,x ) =e}(a, b, c, 2, x 3 , xd} 4, x T = {∅, {b} {d} , {b, , 5{d, , {b, d, d, e}e) , {a, b, d}, Uk = E xk ∈E∈T 1 {a, 0 b,1d, e} 0 , {a, 0 b, c, d} , X} Topogenous matrix 1 1 1 0 0 and consider the colection U = {U1 , . . . , Un }, which is the For the following T orderings we obtain the 0 0 X1 = 0 of0the1elements, minimal basis topological for the space in the sense U isby contained Given a finite space (X, T ),that denote Uk the respective topogenous matrices 1 0 as can be verified by cal1 1 1 in any other for the topology T. minimal openbasis set containing xk : culating the minimal basis0 in 0each 0 case: 0 1 Definition 0.1. Let (X, T ) be a finite topological space (x1 ,x2 , x3 , x4 , x5 ) = (a, b, c, d, e) = minimalEbasis. The topogenous and U = {U1 , . . . , UnU}k its xk ∈E∈T 1 0 1 0 0 to X is the square matrix of matrix TX = [tij ] associated (x1 ,x2 , x3 ,x4 , x5 ) = (b, d, e,a, c) size consider n × n that 1 1 1 0 0 and thesatisfies: colection U = {U1 , . . . , Un }, which is the 01 00 10 01 01 T = X 1 in the sense that U is contained minimal basis for the space 10 01 11 11 11 1 , xi ∈ Uj in any other basistijfor = the topology T . 0 00 01 00 10 TX 2 = 0 0 , in other case 0 0 0 1 1 Definition 0.1. Let (X, T ) be a finite topological space 0 0 0 0 1 and U = {U1 , . . . , Un } its minimal basis. The topogenous Remark 0.2. (Shiraki, 1968) introduce the term topogematrix TX = [tij ] associated to X is the square matrix of (x1 ,x2 , x3 , x4 , x5 ) = (b, d, e, a, c) nousn × matrix denote the transpose matrix of that in be characterized size n thattosatisfies: Topogenous matrices can 1 0 0 1 1 by the following above definition. result. 0 1 1 1 1 1 ,diagram, xi ∈ Uj we represent the minExample 0.3. tIn the next 0 1 Let 0 T0 ij = X2 = 0 1968) Theorem 0.5. T(Shiraki, = [tij ] be the to0 , in other case imal basis for a topology on X = {x1 , x2 , x3 , x4 , x5 , x6 }. 0 0to (X, 0 1T ).1Then T satisfies pogenous matrix associated The minimal open sets are U1 = U5 = {x1 , x5 } , U2 = the following properties, 0for 0all 0i, j,0k ∈1In : , U3 = {x3 } , U41968) = {xintroduce {x4 ,term x6 } and the {x2 , x4 } 0.2. 4 } , U6 = the Remark (Shiraki, topogeassociated is as follows: 1. tij ∈ {0, 1}. nous matrixtopogenous to denote matrix the transpose matrix of that in 2. tii = 1.matrices can be characterized by the following Topogenous 2. tii = 1. above definition. result. 3. tik = tkj = 1 =⇒ tij = 1. 3. tik = tkj = 1 =⇒ tij = 1. Example 0.3. In the next diagram, we represent the minbe the to-n Theorem 0.5. (Shiraki, 1968) Let T = [tij ] size n× imal basis for a topology on X = {x1 , x2 , x3 , x4 , x5 , x6 }. Conversely, if a square matrix T = [tij ] of Conversely, if a associated square matrix T T=). [tThen size n×n pogenous matrix to (X, satisfies ij ] of T The minimal open sets are U1 = U5 = {x1 , x5 } , U2 = satisfies the above properties, T induces a topology on X. satisfies the above properties, the following properties, for allTi,induces j, k ∈ Ina: topology on X. {x2 , x4 } , U3 = {x3 } , U4 = {x4 } , U6 = {x4 , x6 } and the Homeomorphism associated topogenous matrix is as follows: 1. tij ∈ {0, 1}. classes are also described by similarity Homeomorphism classes are also described by similarity via a permutation matrix between topogenous matrices. via a permutation matrix between topogenous matrices. Theorem 0.6. (Shiraki, 1968) Let (X, T ) and (Y, H ) •x4 Theorem 0.6. (Shiraki, 1968) Let (X, T ) and (Y, H ) be finite topological spaces with associated topogenous mabe finite topological spaces with associated topogenous ma•x6 trices TX and TY , respectively. Then (X, T ) and (Y, H ) trices TX and TY , respectively. Then (X, T ) and (Y, H ) are homeomorphic spaces if, and only if, TX and TY are are homeomorphic spaces if, and only if, TX and TY are •x2 similar via a permutation matrix. similar via a permutation matrix. •x5 •x3 Transiting Top(X) •x1 Transiting Top(X) 1 0 0 0 1 0 Using topogenous matrices, we can find a way to transit 0 1 0 0 0 0 Using topogenous matrices, we can find a way to transit through Top(X), the complete lattice of all topologies on 0 0 1 0 0 0 through Top(X), the complete lattice of all topologies on . a fixed set X. Given two topologies T1 and T2 in Top(X), TX = 0 1 0 1 0 1 a fixed set X. Given two topologies T1 and T2 in Top(X), the supremum of them, denoted by T1 ∪T2 , is the topol 1 0 0 0 1 0 the supremum of them, denoted by T1 ∪T2 , is the topology whose open sets are unions of finite intersections of ogy whose open sets are unions of finite intersections of 0 0 0 0 0 1 elements in the collection T1 ∪ T2 . elements in the collection T1 ∪ T2 . Example 0.4. Consider the topological space (X, T ) Proposition 0.7. (Cuevas, 2016) Let (X, T1 ) and Proposition 0.7. (Cuevas, 2016) Let (X, T1 ) and where X = {a, b, c, d, e} and (X, T2 ) be finite topological spaces with minimal basis 128 (X, T ) be finite topological spaces with minimal basis U = 2{U1 , . . . , Un } and V = {V1 , . . . , Vn }, respectively. T = {∅, {b} , {d} , {b, d} , {d, e} , {b, d, e} , {a, b, d}, U = {U1 , . . . , Un } and V = {V1 , . . . , Vn }, respectively. Then, the minimal basis W = {W1 , . . . , Wn } for the space Then, ∗the minimal ∗basis W = {W1 , . . . , Wn } for the space {a, b, d, e} , {a, b, c, d} , X} (X, T ), where T = T ∪ T , satisfies W = U ∩ V
which conta which conta [TX ∗ ]ik = m [TX ∗ ]ik = m
The second The second the minimal the minimal by the above by the above T(X, T1 ∪T2 ) T(X, T1 ∪T2 )
A way to go A way to go is possible u is possible u next result. next result. Corollary Corollary finite topolo finite topolo [TX2 ]ij [TX [TX2 ]ij [TX Proof. The Proof. The lences: lences:
T1 ⊆ T2 ⇐ T1 ⊆ T2 ⇐ ⇐ ⇐
Triangula Triangula
Let (X, T ) b
size n × n size on n ×X. n ology ology on X. by similarity by similarity s matrices. s matrices. and (Y, H ) and (Y, maH) ogenous ogenous and (Y, maH) and (Y, TY H are) and TY are
∗ ∗ Using find a way to Utransit (X, T topogenous ), where Tmatrices, = T1 ∪we T2can , satisfies Wk = k ∩ Vk through Top(X), the complete lattice of all topologies on for all k ∈ In . a fixed set X. Given two topologies T1 and T2 in Top(X), theTop(X), topolthe of them, denoted by T1 ∪T 2 , is in Thesupremum next theorem way to move ahead Rev. Acad. Colomb. Cienc.shows Ex. Fis. a Nat. 41(158):127-136, enero-marzo de 2017 ogy whose open sets are unions of finite intersections of that is, it allows to find the supremum of two topologies. doi: http://dx.doi.org/10.18257/raccefyn.409 T2following . elements in the 1∪ The symbol ∧ iscollection regardedTin the sense: if E and F are n × n matrices, E ∧ F is 2016) the square matrix Proposition 0.7. (Cuevas, Let (X, T1 ) whose and entries satisfy [E ∧ F ] = min {[E] , [F ] }. ij spacesij withij minimal basis (X, T2 ) be finite topological
Let (X, T ) be a finite topological space with minimal basis lences: U = {U1 , . . . , Un } and associated topogenous matrix TX = ]. ⊆Define theTbinary relation on X as follows: [tT ij 1 T2 ⇐⇒ 2 = T1 ∪ T2
The second part of the theorem holds by uniqueness of The minimal second part theorem holds by(1) uniqueness of the basisofforthe a topology, since if is satisfied, thethe minimal for awe topology, since by above basis argument would have TXif∗ (1) = Tis ∧ TX2 = X1satisfied, by(X, T the1above argument would have T thus Twe∗ = T1 ∪ T2 .TX ∗ = TX1 ∧ TX2 = ∪T2 ) and T(X, T1 ∪T2 ) and thus T ∗ = T1 ∪ T2 .
•x2 •x4 •x2 •x4 •x2 •x4 In the example 0.4, it was possible to associate an upper triangular topogenous to the considered since In the example 0.4, it matrix was possible to associatespace an upper such space is T0possible , to as the shows the next Shiraki’s triangular topogenous considered space since In thetopological example 0.4, it matrix was to associate an upper theorem. such topological spacematrix is T0 , to as the shows the next Shiraki’s triangular topogenous considered space since theorem. such topological space is T0 , as shows the next Shiraki’s Theorem theorem. 0.12. (Shiraki, 1968) A finite topological if, and only if, its topogespace (X, T 0.12. ) is T0(Shiraki, Theorem 1968) A associated finite topological is similar via a permutation matrix to an nous matrix T if, and only if, its associated topogespace (X, T ) is T X Theorem 0.12. 0(Shiraki, 1968) A finite topological upper triangular topogenous matrix. similar via a permutation matrix to an nous if, and only if, its associated topogespace matrix (X, T )TX is is T 0 upper triangular topogenous similar viamatrix. a permutation matrix to an nous matrix TX is upper triangular topogenous matrix. A procedure to triangularize a topogenous matrix of a T0 space X is described below. aGiven a topogenous matrix A procedure to triangularize topogenous matrix of a T0 n space X is described below. Given a topogenous matrix A procedure to triangularize a topogenous matrix of a TX = [tij ] define Mk = n tik = |Uk |, for each k ∈ In ;Tif0 Given a topogenous matrix i=1 space described below. =X [tijis ] define = |Uk |, for each k ∈ In ; if TX organize n tik = we them M inkascending order TX = [tij ] define Mk = i=1 tik = |Uk |, for each k ∈ In ; if we organize them in ascending order i=1 we organize themMin ascending order (3) k1 Mk2 · · · Mkn Mk1 Mk2 · · · Mkn (3) Mk1 Mk2 · · · Mk1n 2 · · · n(3) , and consider the permutation σ = k11 k22 · · · knn , and consider the permutation σ = the topogenous matrix PσT TX Pσ is upper k11 k2triangular: · · · knn if 2 and consider the permutation σ = , T would have = 1, P we i > jtopogenous and tσ(i)σ(j) k1xσ(i) ktriangular: · σ(j) · kand the matrix 2 < ·x n if σ TX Pσ is upper T would M=kj1,which the of we have xσ(i) jtopogenous andM tσ(i)σ(j) ki j and tσ(i)σ(j) = 1, we would have xσ(i) < xσ(j) and therefore Mkj Hwhich contradicts the ordering of Cuevas RozoM JL,kSarria i < Zapata Mk , hence tσ(i)σ(j) = 0.
x15 0 1 1 1 0 • 0 0 1 0 1 1 T . P σ T X Pσ = • x4 0 0 0 1 1 1 0Nat.041(158):127-136, 0• 1 Rev. Acad. Colomb. Cienc. Ex. Fis. de 2017 x30 1 enero-marzo 0 0 0 0 0 1 doi: http://dx.doi.org/10.18257/raccefyn.409 • x
Example 0.13. Consider the topological space X, repre•x5 sented by the next Hasse diagram and topogenous matrix:
Remark 0.14. Observe that, in general, the permutation • x1 σ constructed using the relations in (3) is not unique. In example 0.13, we could have used σ = (165)(23) (and, in this case, no other!) to triangularize TX obtaining the upper triangular matrix: 1 1 1 1 1 1 0 1 0 0 1 1 0 0 1 1 1 1 T . P σ T X Pσ = 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 1
•x5 •x1 •x4
•x1 •x3
•x4 •x2 •x2
•x3 •x6 •x6
1 0 0 0 1 0 1 1 0 1 1 0 11 00 10 00 11 00 TX = 11 01 00 11 11 00 1 0 1 0 1 0 0 0 0 0 1 0 TX = 1 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 1 1 First, we find that: M1 = 5, M2 = M3 = 2, M4 = 3, M5 = 6 and M6 = 1. Therefore we obtain the permutation First, we find that: M1 = 5, M2 = M3 = 2, M4 = 3, M5 = 6 and M6 = 1. Therefore 1 2 3 we4 obtain 5 6 the permutation σ= = (165) 6 2 3 4 1 5 1 2 3 4 5 6 σ= = (165) 6 2 3is given 4 1 by5 whose associated matrix
opological d topogeopological trix to an is given by d topoge- whose associated matrix 0 0 0 0 1 0 trix to an 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 . Pσ = 0 1 0 0 0 0 0 0 1 0 0 x of a T0 0 0 . 0 0 0 1 0 0 0 0 0 0 1 Pσ = us matrix x of a T0 0 1 0 0 0 0 1 0 0 0 0 0 k ∈ I ; if 0 0 0 0 0 1 n us matrix 1 0 0 0 0 0 k = xσ(k) , the new ordering of the elek ∈ In ; if If we denote by x ments is represented in the topogenous matrix as follows: If we denote by x k = xσ(k) , the new ordering of the ele(3) ments is represented in the topogenous matrix as follows: • x6 (3) • x6 ·· n , • x5 · · kn ·· n 1• gular: if, x15 1 1 1 1 · · kn • x4 0 1 0 1 1 1 xσ(j) and gular: of if 0 0 1• dering x30 1 1 T . x=4 Pσ TX Pσ• xσ(j) and 0 0 0 1 1 1 • x 2 dering of x30 1 1 0 0 0• • x2 0 0 0 0 0 1 X, repre• x1 s matrix: X, repre Remark 0.14. Observe that, x 1 • 1 1in1 general, 1 1 the 1 permutation s matrix: σ constructed using the not unique. In 0relations 1 0 in 1 (3) 1 is1 have example 0.13, we could used σ = (165)(23) (and, 0 0 1 0 1 1 PσTno TXother!) Pσ = . in this case, triangularize T obtaining the to X 0 0 0 1 1 1 upper triangular matrix: 0 0 0 0 1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 1 1 in general, the Remark 0.14. Observe that, permutation 0 0 1 1 1 1 T .unique. In Pσusing = relations TX Pσthe σ130 constructed is not 0 0 0in (3) 1 1 1 (and, example 0.13, we could have 0 0used 0 σ0 =1 (165)(23) 1 in this case, no other!) to 0triangularize T obtaining the X 0 0 0 0 1
Remark 0.17
Example 0 bering its ve
2
•
For example x5 x8 then the edges w then the firs matrices are
Despite of this lack of uniqueness to choose such permutation σ, the resulting T0 spaces are always homeomorphic (Theorem 0.6), so the topological properties are the same. Remark 0.15. From now on, when (X, T ) is a T0 space, we assume a fixed ordering in the elements of X such that its topogenous matrix TX is upper triangular.
T
Stong matrix Definition 0.16. Given X a T0 space, we define the Stong matrix SX = [sij ] as the square matrix of size n × n that satisfies 1 , xi xj and there is no k with xi < xk < xj sij = 0 , in other case.
S
A simple method to calculate the topogenous matrix TX and the Stong matrix SX of the space X, from the as- Relation sociated Hasse diagram, is described below. Number the matrices vertices so that xi < xj =⇒ i < j, that is, number them from bottom to top ensuring that the topogenous matrix We can reco is upper triangular. For each i = j: matrix and v • tij = 1 if, and only if, there exists a chain whose we see that t of the transi minimum is xi and maximum is xj . • sij = 1 if, and only if, (xi , xj ) is an edge of the diagram. Remark 0.17. tij = 0 ⇒ sij = 0 and sij = 1 ⇒ tij = 1. Example 0.18. Consider the Hasse diagram (left) numbering its vertices as described before (right): an7 edge of•x the • sij = 1 if,• and only•if, (xi , xj ) is •x 8 diagram. •x6 • Remark 0.17. tij = 0 ⇒ sij = 0 and sij = 1 ⇒ tij = 1. •x3 •x4 •x5 • • • Example 0.18. Consider the Hasse diagram (left) numbering its vertices as described before •x (right): •x2 • • 1 For example,• for x5 we•have x5 x5 ,•x x57 x6 , x•x 8 x7 and 5 x5 x8 then the fifth row of TX is [0 0 0 0 1 1 1 1]. Also, • the edges with initial point x1 are (x1 , x•x 3 )6 and (x1 , x4 ), then the first row of SX is [1 0 1 1 0 0 0 0]. The associated •x3 •x4 •x5 • are the •following: • matrices
t unique. In 5)(23) (and, btaining the
•
•
•x3
•
•x4 •x5
Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. 41(158):127-136, enero-marzo de 2017 doi: http://dx.doi.org/10.18257/raccefyn.409 •x2 •x1 • •
For example, for x5 we have x5 x5 , x5 x6 , x5 x7 and x5 x8 then the fifth row of TX is [0 0 0 0 1 1 1 1]. Also, the edges with initial point x1 are (x1 , x3 ) and (x1 , x4 ), then the first row of SX is [1 0 1 1 0 0 0 0]. The associated matrices are the following:
.
ch permutameomorphic re the same.
a T0 space, X such that .
ne the Stong e n × n that
i
•x6
•
< xk < xj
s matrix TX rom the asNumber the umber them nous matrix
TX
1 0 0 0 = 0 0 0 0
SX
1 0 0 0 = 0 0 0 0
0 1 0 0 0 0 0 0
1 0 1 0 0 0 0 0
1 1 0 1 0 0 0 0
0 0 0 0 1 0 0 0
1 1 0 1 1 1 0 0
1 1 0 1 1 1 1 0
1 1 0 1 1 1 0 1
0 1 0 0 0 0 0 0
1 0 1 0 0 0 0 0
1 1 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 1 1 1 0 0
0 0 0 0 0 1 1 0
0 0 0 0 0 1 0 1
remark would imply k < whose j j do tij = 0, else
4.
tij = δ(uij − ujj , 0)
Conflict 5. end for no confli Algorithm: Topogenous matrix from DX 6. end for Referen Input: Matrix DX = [dij ]n×n associated to the function fD of X. Abril J Output: Topogenous matrix TX = [tij ]n×n associated to homotop´ ciones su the space X. sidad Na 1. for i = 1 · · · n got´ a, Co 2. for j = 1 · · · n Alexand (N.S.)2, 3. if i > j do tij = 0, else
Proof. xr ∈ M ⇐⇒ xik xr for all k = 1, . . . , m ⇐⇒ tik r = 1 for all k = 1, . . . , m m ⇐⇒ t ik r = 1 k=1
4.
xr ∈ M ⇐⇒ xr xik for all k = 1, . . . , m ⇐⇒ trik = 1 for all k = 1, . . . , m m ⇐⇒ trik = 1
5.
Cuevas matrices (Trabajo Colombi
tij = δ(dij − dii , 0) end for
6. end for
k=1
Remark 0.25. From proposition 0.24 and above algorithms, we see that if M = {xi1 , . . . , xim } ⊆ X we have:
Therefore |M | =
n r=1
m
k=1
ti k r
and |M | =
n r=1
m
k=1
trik
.
Proposition 0.22 allows to obtain the matrices UX and DX from the topogenous matrix TX . Consider now the reverse process of obtaining the topogenous matrix from UX and from DX . In our matrix language, (5) and (6) are equivalent to having tij = 1 ⇐⇒ ujj = uij ⇐⇒ dii = dij Therefore, we can reconstruct the topogenous matrix by using the following algorithms: Algorithm: Topogenous matrix from UX Input: Matrix UX = [uij ]n×n associated to the function f of X.
fU (M ) = n −
n
fD (M ) = n −
n
r=1
r=1
m
k=1
m
k=1
δ(fU (xr ) − fU (xik , xr ), 0)
δ(fD (xik ) − fD (xik , xr ), 0)
Therefore, functions fU and fD can be calculated for any subset M of the space X, from the functions values in the subsets of cardinality one and two through the above explicit formulas.
Conclusions
135
Krishna on a fini
Sarria, entre los funcione 31-50.
Shiraki, Fac. Sci
Stong R Amer. M
Therefore, functions fU and fD can calculated for any subset M of the space X, from thebefunctions values in fD of X. subset M of the space X, from the functions values in the subsets of cardinality one and two through the above [tijthrough ]n×n associated to Output: Topogenous matrix TX = the subsets of cardinality one and two the above explicit formulas. the space X. explicit formulas. Cuevas Rozo JL, Sarria Zapata H 1. for i = 1 · · · n
Conclusions 2. for j = 1 · · · n Conclusions
3. if i > j do tij = 0, else We have seen how the considered matrices make easier the We4.have seen the matrices make easier the = δ(d − dspaces tijhow topological study in finite we use submodular ij considered ii , 0) when topological study in finite spaces when we use functions, achieving to eliminate the need to submodular draw Hasse 5. end for functions, to eliminate the needinto draw Hasse diagrams toachieving find its values as was worked (Abril, 2015) diagrams to find its values as was worked in (Abril, 2015) 6. end for and (Sarria et al., 2014). In future works, it can be and (Sarria et al., 2014). In future works, it can be studied complexity of the shown algorithms and try to find studiedtopological complexityconcepts of the shown algorithms try to find other which could beand characterized other topological concepts which could from this matrix point of view. Remark 0.25. From proposition 0.24 be andcharacterized above algofrom this matrix point of view. rithms, we that if The M =authors {xi1 , . . declare . , xim } that ⊆ X they we have: Conflict of see interest. have Conflict no conflictofofinterest. interest. The authors declare that they have no conflict of interest. m n References function References δ(fU (xr ) − fU (xik , xr ), 0) fU (M ) = n − function r=1 k=1 Abril J. A. (2015). Una aproximaci´ on a la noci´ on de J.ıa A. (2015). Una aproximaci´ on a desde la noci´ de homotop´ entre espacios topol´ ogicos finitos lasonfunciated to Abril ıa entre espacios topol´ o gicos finitos desde las funciated to homotop´ ciones submodulares (Trabajo Final de Maestr´ıa). Univer m ciones submodulares de Maestr´ ıa). Univern (Trabajo Final sidad Nacional de Colombia. Facultad de Ciencias. Bosidad Nacional Facultad Ciencias. (M ) = n de − Colombia. δ(fD (xik ) − fde D (x ik , xr ), 0) Bogot´ af, DColombia. got´ a, Colombia. r=1 k=1 Alexandroff P. (1937). Diskrete R¨ aume. Mat. Sb. Alexandroff P. (1937). Diskrete R¨ aume. Mat. Sb. (N.S.)2, 501-518. Therefore, functions fU and fD can be calculated for any (N.S.)2, 501-518. subset M of the X, from the functions values yin Cuevas Rozo J. space L. (2016). Funciones submodulares Cuevas Rozo J. L. (2016). Funciones submodulares y the subsets of estudio cardinality oneespacios and two topol´ through thefinitos above matrices en el de los ogicos matrices en el estudio de los espacios topol´ o gicos finitos explicit formulas. (Trabajo Final de Maestr´ıa). Universidad Nacional de (Trabajo de Maestr´ ıa). Universidad Nacional de Colombia.Final Facultad de Ciencias. Bogot´ a, Colombia. Colombia. Facultad de Ciencias. Bogot´ a, Colombia. Krishnamurthy V. (1966). On the number of topologies Krishnamurthy V. (1966). the number of topologies Conclusions on a finite set. Amer. Math. On Montly 73, 154-157. ve algo- on a finite set. Amer. Math. Montly 73, 154-157. algo- Sarria, H., Roa L. & Varela R. (2014). Conexiones evehave: We have seenRoa how L. the& considered matrices make easier the Sarria, Varela R. (2014). Conexiones e have: entre los H., espacios topol´ ogicos finitos, las fd relaciones y las topological studytopol´ in finite spaces when we use submodular entre los espacios o gicos finitos, las fd relaciones y las funciones submodulares. Bolet´ın de Matem´ aticas, 21(1), functions, achieving to eliminate the need to draw Hasse funciones submodulares. Bolet´ ın de Matem´ a ticas, 21(1), 31-50. diagrams to find its values as was worked in (Abril, 2015) 31-50. , 0) and (Sarria et al., On 2014). future works, it can be Shiraki, M. (1968). finiteIntopological spaces. Rep. , 0) Shiraki, M. (1968).ofUniv., On finite topological spaces. Rep. studied the shown algorithms and try to find Fac. Sci.complexity Kagoshima 1, 1-8. Fac. Kagoshima Univ., 1, 1-8. could be characterized otherSci. topological concepts which Stong R. E. (1966). Finite topological spaces. Trans. from this pointFinite of view. Stong R. matrix E. Soc., (1966). topological spaces. Trans. Amer. Math. 123(2), 325–340. Amer. Math. Soc., 123(2), 325–340. ), 0) ), 0)
d for any d for any values in values in he above he above
asier the asier the modular modular aw Hasse aw il, Hasse 2015) 2015) til,can be ty can be to find y to find acterized acterized 136
Abril J. A. (2015). Una aproximaci´ on a la noci´ on de homotop´ıa entre espacios topol´ ogicos finitos desde las funciones submodulares (Trabajo Final de Maestr´ıa). UniverRev. Acad. Colomb. Cienc. Ex. Fis. Nat. 41(158):127-136, enero-marzo de 2017 sidad Nacional de Colombia. Facultad de Ciencias. Bodoi: http://dx.doi.org/10.18257/raccefyn.409 got´a, Colombia. Alexandroff P. (1937). (N.S.)2, 501-518.
Diskrete R¨aume.
Mat.
Sb.
Cuevas Rozo J. L. (2016). Funciones submodulares y matrices en el estudio de los espacios topol´ ogicos finitos (Trabajo Final de Maestr´ıa). Universidad Nacional de Colombia. Facultad de Ciencias. Bogot´a, Colombia. Krishnamurthy V. (1966). On the number of topologies on a finite set. Amer. Math. Montly 73, 154-157. Sarria, H., Roa L. & Varela R. (2014). Conexiones entre los espacios topol´ogicos finitos, las fd relaciones y las funciones submodulares. Bolet´ın de Matem´ aticas, 21(1), 31-50. Shiraki, M. (1968). On finite topological spaces. Rep. Fac. Sci. Kagoshima Univ., 1, 1-8. Stong R. E. (1966). Finite topological spaces. Trans. Amer. Math. Soc., 123(2), 325–340.
