Some New Types of Transitivity and Minimality

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1 Ope Access Library Joural 2018, Volume 5, e4852 ISSN Olie: ISSN Prit: Some New Types of Trasitivity ad Miimality Mohammed Nokhas Murad Kaki Math Departmet, College of Sciece, Uiversity of Sulaymaiyah, Sulaymaiyah, Iraqi How to cite this paper: Kaki, MNM (2018) Some New Types of Trasitivity ad Miimality Ope Access Library Joural, 5: e Received: August 20, 2018 Accepted: November 20, 2018 Published: November 23, 2018 Copyright 2018 by author ad Ope Access Library Ic This work is licesed uder the Creative Commos Attributio Iteratioal Licese (CC BY 40) Ope Access Abstract I this paper, we itroduce ad study the relatioship betwee two differet otios of trasitive maps, amely topological -trasitive maps, topological -trasitive maps ad ivestigate some of their properties i two topological spaces (X, τ ) ad (X, τ ), τ deotes the -topology (resp τ deotes the -topology) of a give topological space (X, τ) The two otios are defied by usig the cocepts of -irresolute map ad -irresolute map respectively Also, we defie ad study the relatioship betwee two types of miimal mappigs, amely, -miimal mappig ad -miimal mappig, The mai results are the followig propositios: 1) Every topologically -trasitive map is trasitive map, but the coverse is ot ecessarily true 2) Every topologically -miimal map is miimal map, but the coverse is ot ecessarily true 3) The coverse of (1) ad (2) is ot ecessarily true uless every owhere dese set i ( X, τ ) is closed 4) Also, if every -ope set is locally closed the every trasitive map implies topological -trasitive Subject Areas Mathematical Aalysis Keywords Topologically -Trasitive, -Irresolute, -Trasitive, -Dese 1 Itroductio The cocept of topological trasitivity goes back to G D Birkhoff who itroduced it i 1920 (for flows) This article will cocetrate o topological trasitivity of dyamical systems give by cotiuous mappigs i metric spaces A dyamical system is a rule for time evolutio o a state space Ituitively, a topologically trasitive dyamical system has poits which evetually move uder iteratio from oe arbitrarily small ope set to ay other Cosequetly, such a DOI: /oalib Nov 23, Ope Access Library Joural

2 M N M Kaki dyamical system caot be decomposed ito two disjoit sets with oempty iteriors which do ot iteract uder the trasformatio Birkhoff was oe of the most importat leaders i America mathematics i his geeratio, ad durig his time he was cosidered by may to be the preemiet America mathematicia Recetly there has bee some iterest i the otio of a locally closed subset of a topological space Accordig to Bourbaki [1] a subset S of a space ( X, τ ) is called locally closed if it is the itersectio of a ope set ad a closed set Gaster ad Reilly used locally closed sets i [2] ad [3] to defie the cocept of LC-cotiuity ie a fuctio f :( X, τ) ( X, σ) is LC-cotiuous if the iverse with respect to f of ay ope set i Y is closed i X The study of semi ope sets ad semi cotiuity i topological spaces was iitiated by Levie [4] Bhattacharya ad Lahiri [5] itroduced the cocept of semi geeralized closed sets i topological spaces aalogous to geeralized closed sets which were itroduced by Levie [6] Throughout this paper, the word space will mea topological space The collectios of semi-ope, semi-closed sets ad -sets i ( X, τ ) will be deoted by SO ( X, τ ), SC ( X, τ ) ad τ respectively Njastad [7] has show that τ is a topology o X with the followig properties: τ τ, ( ) τ = τ ad S τ if ad oly if S = U \ N where U τ ad N is owhere dese ( i e It ( Cl ( N )) = ϕ ) i ( X, τ ) Hece τ = τ if ad oly if every owhere dese (wd) set i ( X, τ ) is closed, therefore every trasitive map implies -trasitive Also if every -ope set is locally closed the every trasitive map implies -trasitive ad this structure also occurs if ( X, τ ) is locally compact ad Hausdorff ([8], p 140, Ex B) ad every -ope set is locally compact, the every -ope set is locally closed Clearly every -set is semi-ope ad every wd set i ( X, τ ) is semi-closed Adrijevic [9] has ob- SO X, τ = SO X, τ, ad that N X X, τ if served that ( ) ( ) is wd i ( ) ad oly if N is wd i ( X, τ ) I 1943, Fomi [10] itroduced the otio of -cotiuous maps The otios of -ope sets, -closed sets ad -closure where itroduced by Velicko [11] for the purpose of studyig the importat class of H-closed spaces i terms of arbitrary fiber-bases Dickma ad Porter [12] [13], Joseph [14] ad Log ad Herrigto [15] cotiued the work of Velicko We itroduce the otios of -type trasitive maps, -miimal maps ad show that some of their properties are aalogous to those for topologically trasitive maps Also, we give some additioal properties of -irresolute maps We deote the iterior ad the closure of a subset A of X by It(A) ad Cl(A), respectively By a space X, we mea a topological space ( X, τ ) A poit x X is called a -adheret poit of A [11], if A Cl ( V ) φ for every ope set V cotaiig x The set of all -adheret poits of a subset A of X is called the -closure of A ad is deoted by Cl A subset A of X is called -closed if A = Cl Dotchev ad Maki [14] have show that if A ad B are subsets of a space ( X, τ ), the Cl ( A B) = Cl Cl ad Cl ( A B) = Cl Cl Note al- DOI: /oalib Ope Access Library Joural

3 M N M Kaki so that the -closure of a give set eed ot be a -closed set But it is always closed Dickma ad Porter [12] proved that a compact subspace of a Hausdorff space is -closed Moreover, they showed that a -closed subspace of a Hausdorff space is closed Jakovic [16] proved that a space ( X, τ ) is Hausdorff if ad oly if every compact set is -closed The complemet of a -closed set is called a -ope set The family of all -ope sets forms a topology o X ad is deoted by τ or topology This topology is coarser tha τ I geeral, Cl will ot be the closure of A with respect to ( X, τ ) It is easily see that oe always has A Cl Cl Cl A δ where A deotes the closure of A with respect to ( X, τ ) It is also obvious that a set A is X, X, τ is -closed i (, ) X τ if ad oly if it is closed i ( τ ) The space ( ) called sometimes the semi regularizatio of ( X, τ ) A fuctio f : X Y is closure cotiuous [17] (-cotiuous) at x X if give ay ope set V i Y cotaiig f(x), there exists a ope set U i X cotaiig x such that f ( Cl ( U )) Cl ( V ) [17] I this paper, we will defie ew classes of topological trasitive maps called -type trasitive, -type trasitive ad ew classes of -miimal maps -miimal maps We have show that every -trasitive map is a -type trasitive map, but the coverse ot ecessarily true ad that every -miimal map is a -miimal map, but the coverse ot ecessarily true we will also study some of their properties 2 Prelimiaries ad Defiitios I this sectio, we recall some of the basic defiitios Let X be a space ad A X The itersectio (resp closure) of A is deoted by It(A) (resp Cl(A) Defiitio 21 [4] A subset A of a topological space X will be termed semi-ope (writte SO) if ad oly if there exists a ope set U such that U A Cl ( U ) Defiitio 22 [5] Let A be a subset of a space X the semi closure of A defied as the itersectio of all semi-closed sets cotaiig A is deoted by scla Defiitio 23 [10] Let ( X, τ ) be a topological space ad a operator from τ to Ƥ(X) ie : τ Ƥ(X), where Ƥ(X) is a power set of X We say that is a operator associated with τ if U ( U) for all U τ Defiitio 24 [18] Let ( X, τ ) be a topological space ad a operator associated with τ A subset A of X is said to be -ope if for each x X there exists τ, SO(τ), re- a ope set U cotaiig x such that ( U) all -ope, semi-ope sets i the topological space ( X, τ ) by spectively We the have τ τ SO ( τ) A Let us deote the collectio of A subset B of X is said to be -closed [7] if its complemet is -ope Defiitio 25 [10] Let ( X, τ ) be a space A operator is said to be regular if, for every ope eighborhoods U ad V of each x X, there exists a eighborhood W of x such that ( W) ( U) ( V) Note that the family X, τ always forms a topology o X, whe is cosidered to be regular fier tha τ τ of -ope sets i ( ) DOI: /oalib Ope Access Library Joural

4 M N M Kaki Theorem 26 [20] For subsets A, B of a space X, the followig statemets hold: 1) D D where ( ) 2) If A B, the A B D A is the derived set of A 3) D D = D ( A B) ad D ( A B) D D Note that the family τ of -ope sets i (, ) o X deoted -topology ad that -topology coarser tha τ X τ always forms a topology Defiitio 27 [19] Let A be a subset of a space X A poit x is said to be a -limit poit of A if for each -ope U cotaiig x, U ( A\ x) φ all -limit poits of A is called the -derived set of A ad is deoted by D The set of Defiitio 28 [19] For subsets A ad B of a space X, the followig statemets hold true: 1) D D 2) If A B the D D 3) D D D ( A B) 4) D ( A D ) A D where D(A) is the derived set of A Defiitio 29 [18] The poit x X is i the -closure of a set A X if ( U) A ϕ itersectio of all -closed sets cotaiig A ad is deoted by Cl Remark 210 For ay subset A of the space X, A Cl Cl Defiitio 211 [18] Let (, ), for each ope set U cotaiig x The -closure of a set A is the X τ be a topological space We say that a subset A of X is -compact if for every -ope coverig Π of A there exists a fiite sub-collectio { C C C },,, of Π such that 1 2 A Ci Properties of -compact spaces have bee ivestigated by Rosa, E etc ad Kasahara, S [18] The followig results were give by Rosas, E etc [18] Theorem 212 Let ( X, τ ) be a topological space ad a operator associated with τ A X ad K A If A is -compact ad K is -closed the K is -compact Theorem 213 Let ( X, τ ) be a topological space ad be a regular operator o τ If X is -T 2 (see Rosa, E etc ad Kasahara, S) [10] ad K -compact the K is -closed i= 1 X is Defiitio 214 [18] The itersectio of all -closed sets cotaiig A is called the -closure of A, deoted by Cl Remark 215 For ay subset A of the space X, A Cl Cl Lemma 216 For subsets A ad A (i I) of a space ( X, τ ) hold: 1) A Cl 2) Cl is closed Cl ( Cl ) Cl 3) If A B the Cl Cl 4) Cl ( ( Ai : i I )) ( Cl : i I ) 5) Cl ( ( Ai : i I )) = ( Cl : i I ) i =, the followig Lemma 217 The collectio of -compact subsets of X is closed uder fiite DOI: /oalib Ope Access Library Joural

5 M N M Kaki uios If is a regular operator ad X is a -T 2 space the it is closed uder arbitrary itersectio Defiitio 218 Let (X, τ) be a topological ay space, A subset of X, The it = { U : U is -ope ad U A} Remark 219 A subset A is -ope if ad oly if it = A Proof: The proof is obvious from the defiitio Defiitio 220 Let (X, τ) ad (Y, σ) be two topological spaces, a map f : X Y is said to be cotiuous if for each ope set H of Y, f ( H) is -ope i X Theorem 221 [19] For ay subset A of a space X, Cl = A Cl Theorem 222 [19] For subsets A, B of a space X, the followig statemets are true: 1) it is the largest ope cotaied i A 2) it ( it ) = it 3) If A B the it it 4) it it it ( A B) 5) it it it ( A B) Lemma 223 [7] For ay -ope set A ad ay -closed set C, we have: 1) Cl = Cl 2) it ( C) = it ( C) 3) it ( Cl ) = it ( Cl ) Remark 224 [19] It is ot always true that every -ope set is a ope set, as show i the followig example: Example 225 Let X= { abcd,,, } with topology τ = { φ, { ab, }, X} Hece ( τ) = { φ, { cd, },{ bcd,, },{ acd,, }, X} So {b, c, d} is -ope but ot ope Theorem 226 [20] For subsets A, B of a topological space X, the followig statemets are true: 1) It is the uio of all ope sets of X whose closures are cotaied i A 2) A is -ope It = A 3) It ( It ) It 4) X \ It = Cl ( X \ A) 5) X \ Cl = It ( X \ A) 6) If A B the It It 7) It It It ( A B) It A It B = It A B 8) ( ) ( ) ( ) 3 Trasitive ad Miimal Systems Topological trasitivity is a global characteristic of dyamical systems By a dyamical system ( X, f ) [21] we mea a topological space X together with a cotiuous map f : X X The space X is sometimes called the phase space of the system A set A X f A A is called f-iveriat if ( ) DOI: /oalib Ope Access Library Joural

6 M N M Kaki A dyamical system ( X, f ) is called miimal if X does ot cotai ay o-empty, proper, closed f-iveriat subset I such a case we also say that the map f itself is miimal Thus, oe caot simplify the study of the dyamics of a miimal system by fidig its otrivial closed subsystems ad studyig first the dyamics restricted to them 2 Give a poit x i a system ( X, f ), O ( ) {, ( ), ( ), f x = x f x f x } deotes its orbit (by a orbit we mea a forward orbit eve if f is a homeomorphism) ad ω f ( x) deotes its ω-limit set, ie the set of limit poits of the sequece 2 x, f ( x), f ( x), The followig coditios are equivalet: ( X, f ) is miimal, every orbit is dese i X, ω f ( x) = X for every x X A miimal map f is ecessarily surjective if X is assumed to be Hausdorff ad compact Now, we will study the Existece of miimal sets Give a dyamical system ( X, f ), a set A X is called a miimal set if it is o-empty, closed ad ivariat ad if o proper subset of A has these three properties So, A X is a X, f is miimal set if ad oly if (, ) A f A is a miimal system A system ( ) miimal if ad oly if X is a miimal set i ( X, f ) The basic fact discovered by G D Birkhoff is that i ay compact system ( X, f ) there are miimal sets This follows immediately from the Zor s lemma Sice ay orbit closure is ivariat, we get that ay compact orbit closure cotais a miimal set This is how compact miimal sets may appear i o-compact spaces Two miimal sets i ( X, f ) either are disjoit or coicide A miimal set A is strogly f-iveriat, ie f = A Provided it is compact Hausdorff Let ( X, f ) be a topological system, ad f : X X r-homeomorphism of X oto itself For A ad B subsets of X, we let N( AB, ) = { Z : f B φ} We write N( AB, ) = N( xb, ) for a sigleto A= { x} thus N( xb, ) = { Z : f ( x) B} For a poit x X we write Of ( x) = { f ( x) : Z } for the orbit of x ad Cl ( Of ( x) ) for the -closure of Of ( x ) We say that the topological system ( X, f ) is -type poit trasitive if there is a poit x X with O ( x ) -dese Such a poit is called -type trasitive We say that the topological system ( X, f ) is topologically -type trasitive (or just -type trasitive) if the set NUV (, ) is oempty for every pair U ad V of oempty -ope subsets of X Topologically -Trasitive Maps I [22], we itroduced ad defied a ew class of trasitive maps that are called topologically -trasitive maps o a topological space (X, τ), ad we studied some of their properties ad proved some results associated with these ew defiitios We also defied ad itroduced a ew class of -miimal maps I this f DOI: /oalib Ope Access Library Joural

7 M N M Kaki paper we discuss the relatioship betwee topologically -trasitive maps ad -trasitive maps O the other had, we discuss the relatioship betwee -miimal ad -miimal i dyamical systems Defiitio 311 Let (X, τ) be a topological space A subset A of X is called -dese i X if Cl = X Remark 312 Ay -dese subset i X itersects ay -ope set i X Proof: Let A be a -dese subset i X, the by defiitio, Cl X =, ad let U be a o-empty -ope set i X Suppose that A U = φ Therefore c c B = U is -closed ad A U = B So Cl Cl, ie Cl B, but Cl = X, so X B, this cotradicts that U ϕ Defiitio 313 [23] A map f : X Y is called -irresolute if for every -ope set H of Y, f ( H) is -ope i X Example 314 [22] Let (X, τ) be a topological space such that X= { abcd,,, } ad τ = { φ, X, { ab, },{ b} } We have the set of all -ope sets is ( X, τ) = { φ, X, { b},{ ab, },{ bc, },{ bd, },{ abc,, },{ abd,, }} ad the set of all -closed sets is C( X, τ) = { φ, X, { c, d},{ a, c, d},{ a, d},{ a, c},{ d},{ c} } The defie the map f : X X as follows f ( a) = a, f ( b) = b, f ( c) = d, f ( d) = c, we have f is -irresolute because {b} is -ope ad f ({ b} ) = { b} is -ope {a, b} is -ope ad f ({ ab, }) = { ab, } is -ope {b, c} is -ope ad f ({ bc, }) = { bd, } is -ope {a, b, c} is -ope ad f ({ abc,, }) = { abd,, } is -ope {a, b, d} is -ope ad f ({ abd,, }) = { abc,, } is -ope so f is -irresolute Defiitio 315 A subset A of a topological space (X, τ) is said to be owhere -dese, if its -closure has a empty -iterior, that is, it ( Cl ) = φ Defiitio 316 [22] Let (X, τ) be a topological space, f : X X be -irresolute map the f is said to be topological -trasitive if every pair of o-empty -ope sets U ad V i X there is a positive iteger such that f ( U) V φ I the forgoig example 314: we have f is -trasitive because b belogs to ay o-empty -ope set V ad also belogs to f(u) for ay -ope set it meas that f ( U) V φ so f is -trasitive Example 317 Let (X, τ) be a topological space such that X= { abc,, } ad τ = { φ, { a}, X} The the set of all -ope sets is τ = { φ, { a},{ ab, },{ ac, }, X} Defie f : X X as follows f ( a) = b, f ( b) = b, f ( c) = c Clearly f is cotiuous because {a} is ope ad f ({ a} ) = φ is ope Note that f is trasitive because f ({ a} ) = { b} implies that f ({ a} ) { b} φ But f is ot -trasitive because for each i N, f ({ a} ) { ac, } = φ sice f ({ a} ) = { b} for every N, ad { b} { ac, } = φ So we have f is ot -trasitive, so we show that trasitivity ot implies -trasitivity Defiitio 318 Let (X, τ) be a topological space A subset A of X is called -dese i X if Cl = X Remark 319 Ay -dese subset i X itersects ay -ope set i X Proof: Let A be a -dese subset i X, the by defiitio, Cl X =, ad let U be a o-empty -ope set i X Suppose that A U = φ Therefore DOI: /oalib Ope Access Library Joural

8 M N M Kaki B c = U is -closed because B is the complemet of -ope ad So Cl Cl, ie Cl B, but ( ) Cl A = X, so X B, this cotradicts that U ϕ c A U = B Defiitio 3110 [24] A fuctio f : X X is called -irresolute if the iverse image of each -ope set is a -ope set i X Defiitio 3111 A subset A of a topological space (X, τ) is said to be owhere -dese, if its -closure has a empty -iterior, that is, ( ( )) = φ Defiitio 3112 [25] Let (, ) it Cl A X τ be a topological space, ad f : X X -irresolute) map, the f is said to be topologically -type trasitive map if for every pair of -ope sets U ad V i X there is a positive iteger such that f ( U) V φ Theorem 3113 every theta-type trasitive map implies trasitive ap if (X, τ) is regular Note that a space (X, τ) is regular if ad oly if τ = τ [26] Theorem 3114 [22] Let (X, τ) be a topological space ad f : X X be -irresolute map The the followig statemets are equivalet: 1) f is topological -trasitive map 2) For every oempty -ope set U i X, f ( U) is -dese i X = 0 3) For every oempty -ope set U i X, f ( U) 4) If B X is owhere -dese X is -dese i X = 0 is -closed ad B is f-ivariat ie ( ) 5) If U is -ope ad ( ) f B B the B = X or B f U U the U is either empty set or -dese i Theorem 3115 [25] Let (X, τ) be a topological space ad f : X X be -irresolute map The the followig statemets are equivalet: 1) f is -type trasitive map is -dese i X, with D is -ope set i X 2) f ( D) = 0 is -dese i X with D is -ope set i X 3) f ( D) = 0 4) If B X -dese 5) If ( ) is -closed ad ( ) f B B the B = X or B is owhere f D D ad D is -ope i X the D = ϕ or D is -dese i X, we have to prove th theorem 4 -Miimal Fuctios We itroduced a ew defiitio o -miimal [22] (resp -miimal [25]) maps ad we studied some ew theorems associated with these defiitios Give a topological space X, we ask whether there exists -irresolute (resp { : 0} -irresolute) map o X such that the set ( ) ad deoted by O ( ) f f x, called the orbit of x x, is -dese(resp -dese) i X for each x X A partial DOI: /oalib Ope Access Library Joural

9 M N M Kaki aswer will be give i this sectio Let us begi with a ew defiitio Defiitio 41 (-miimal) Let X be a topological space ad f be -irresolute map o X with -regular operator associated with the topology o X The the dyamical system (X, f) is called -miimal system (or f is called -miimal map o X) if oe of the three equivalet coditios [22] hold: 1) The orbit of each poit of X is -dese i X 2) Cl ( Of ( x) ) = X for each x X 3) Give x X ad a oempty -ope U i X, there exists N such that f ( x) U Theorem 42 [22] For (, ) E X f the followig statemets are equivalet: 1) f is a -miimal map 2) If E is a -closed subset of X with f ( E) E, we say E is ivariat The = φ or E = X f U = X 3) If U is a oempty -ope subset of X, the ( ) 5 Coclusios We have followig propositios: 1) Every topologically -trasitive map is trasitive map, but the coverse is ot ecessarily true 2) Every topologically -miimal map is miimal map, but the coverse is ot ecessarily true 3) The coverse of (1) ad (2) is ot ecessarily true uless every owhere dese set i ( X, τ ) is closed Also, if every -ope set is locally closed the every trasitive map implies topological -trasitive Coflicts of Iterest The author declares o coflicts of iterest regardig the publicatio of this paper Refereces [1] Bourbaki, N (1966) Geeral Topology Part 1, Addiso Wesley, Readig, MA [2] Gaster, M ad Reilly, IL (1990) A Decompositio of Cotiuity Acta Mathematica Hugarica, 56, [3] Gaster, M ad Reilly, IL (1989) Locally Closed Sets ad LC-Cotiuous Fuctios Iteratioal Joural of Mathematics ad Mathematical Scieces, 12, [4] Levie, N (1963) Semi-Ope Sets ad Semi-Cotiuity i Topological Spaces The America Mathematical Mothly, 70, [5] Bhattacharya, P ad Lahiri, KB (1987) Semi-Geeralized Closed Sets i Topology Idia Joural of Mathematics, 29, [6] Levie, N (1970) Geeralized Closed Sets i Topology Redicoti del Circolo = 0 DOI: /oalib Ope Access Library Joural

10 M N M Kaki Matematico di Palermo, 19, [7] Ogata, N (1965) O Some Classes of Nearly Ope Sets Pacific Joural of Mathematics, 15, [8] Egelkig, R (1968) Outlie of Geeral Topology North Hollad Publishig Compay, Amsterdam [9] Adrijević, D (1994) Some Properties of the Topology of -Sets Matematički Vesik, 36, 1-9 [10] Rosas, E ad Vielia, J (1998) Operator-Compact ad Operator-Coected Spaces Scietific Mathematica, 1, [11] Velicko, NV (1966) H-Closed Topological Spaces America Mathematical Society Traslatios, 78, [12] Dickma, RF ad Porter, JR (1975) -Closed Subsets of Hausdorff Spaces Pacific Joural of Mathematics, 59, [13] Dickma Jr, RF ad Porter, JR (1977) -Perfect ad -Absolutely Closed Fuctios Illiois Joural of Mathematics, 21, [14] Dotchev, J ad Maki, H (1998) Groups of -Geeralized Homeomorphisms ad the Digital Lie Topology ad Its Applicatios, 20, 1-16 [15] Log, PE ad Herrigto, LL (1982) The τ -Topology ad Faitly Cotiuous Fuctios Kyugpook Mathematical Joural, 22, 7-14 [16] Jakovic, DS (1986) -Regular Spaces Iteratioal Joural of Mathematics ad Mathematical Scieces, 8, [17] Saleh, M (2003) O -Cotiuity ad Strog -Cotiuity Applied Mathematics E-Notes, 3, [18] Kasahara, S (1979) Operatio-Compact Spaces Mathematica Japoica, 24, [19] Caldas, M (2003) A Note o Some Applicatios of -Ope Sets UMMS, 2, [20] Caldas, M, Jafari, S ad Kovar, MM (2004) Some Properties of -Ope Sets Divulgacioes Matematicas, 12, [21] [22] Kaki, MNM (2012) Topologically Trasitive Maps ad Miimal Systems Geeral Mathematics Notes, 10, [23] Maheshwari, NS ad Thakur, SS (1980) O -Irresolute Mappigs Tamkag Joural of Mathematics, 11, [24] Khedr, FH ad Noiri, T (1986) O -Irresolute Fuctios Idia Joural of Mathematics, 28, [25] Kaki, MNM (2012) Itroductio to -Type Trasitive Maps o Topological Spaces Iteratioal Joural of Basic & Applied Scieces, 12, [26] Joseph, JE (1979) -Closure ad -Subclosed Graphs The Mathematical Chroicle, 8, DOI: /oalib Ope Access Library Joural