EXT /09/1998

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1 Available at: off IC/98/128 United ations Educational Scientic and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM ITERATIOAL CETRE FOR THEORETICAL PHYSICS EXT /09/1998 TAUTESS AD APPLICATIOS OF THE ALEXADER-SPAIER COHOMOLOGY OF -TYPES Abd El-Sattar A. Dabbour 1 Department of Mathematics, Faculty of Science, Ain-Shams University, Abbasia, Cairo, Egypt and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy and Rola A. Hijazi Department of Mathematics, Faculty of Science, ing Abdulaziz University, P.O. Box 14466, Jeddah 21424, Saudi Arabia. Abstract The aim of the present work is centered around the tautness property for the two -types of Alexander-Spanier cohomology given by the authors. Aversion of the continuity property is proved, and some applications are given. MIRAMARE { TRIESTE September Regular Associate of the Abdus Salam ICTP. (c/o): sherif@sunet.shams.eg

2 0 Introduction It is well-known that in the Alexander-Spanier cohomology theory [17], [18] or in the isomorphic theory of Cech [9], if the coecient group G is topological then either the theory does not take into account the topology on G [9], [18], or considers only the case when G is compact to obtain a compact cohomology [5], [8]. Continuous cohomology naturally arises when the coecient group of a cohomology theory is topological [6],[7],[11]. The partially continuous Alexander-Spanier cohomology theory [14] can be considered as a variant of the continuous cohomology of a space with two topologies in the sense of Bott-Haeiger [15]; also it is isomorphic to the continuous cohomology of a simplicial space dened by Brown-Szczarba [6]. The idea of -groups [1],[2] where is a locally-nite simplicial complex, is used to introduce the -types of Alexander-Spanier cohomology with coecients in a pair (G; G 0 ) of topological abelian groups [3],[4]; namely, -Alexander-Spanier and partially continuous - Alexander-Spanier cohomologies H, H ~. It is proved that these -types satised the seven Eilenberg-Steenrod axioms [9]; the excision axiom for the second -type is veried for compact Hausdor spaces when (G; G 0 ) are absolutely retract. Therefore the uniueness theorem of the cohomology theory on the category of compact polyhedral pairs [9], asserts that our Alexander-Spanier -types over a pair of absolute retract coecient abelian groups are naturally isomorphic. The present work is centered around the tautness property for the Alexander-Spanier - types cohomology. Roughly speaking, we prove that the -Alexander-Spanier cohomology of a closed subset in a paracompact space is isomorphic to the direct limit of the -Alexander-Spanier cohomology of its neighborhoods; and that the partially continuous -Alexander-Spanier cohomology of a neighborhood retract closed subspace of a Hausdor space is isomorphic to the direct limit of the partially continuous -Alexander-Spanier cohomology of its neighborhoods. Also a version of the continuity property is proved. Moreover, we study some application of the -type cohomologies. 1 Alexander-Spanier Cohomology of Types Here we mention the notations which will be used throughout the present work [3],[4]. For an object (X; A) of the category Q of the pairs of topological spaces and their continuous maps, denote by (X; A)[~(X; A)] the set of the pairs =(; 0 ), where is an open covering of X and 0 is a subcollection of covering A[ 0 = \ A]; it is directed with respect to the renement relation <, i.e. < and 0 < 0 [9]. Denote by C ( ) ( ~ X) the group of the functions ' : ~ X ( )+1! G, where is a simplex in, () = + dim, 0, and ~ X denotes either a space X or 2 (X). Let C ( ) ( ~ X) be the subgroup of the direct product Q 2 C( ) ( ~ X) consisting of such ' = f' g for which the condition (k) is satised, which states that there is a conite subset (') of, i.e. (') is nite, such that (' ) 1 (G 0 )= ~ X ()+1, 2

3 8 2 ('). The coboundary : C ( ~ X)! C +1 ( ~ X)is ( ') = X ()+1 i=1 X ( 1) i ' (( )+1) p i +( 1) ( )+1 2st( ) [ : ]' ; where st() =f2: is (dim 1){face of g, p ( ) i : X +1! X is the projection dened by: if ^ti is the -tuple consisting of t =(x 0 ;:::;x ) 2 X +1 with x i omitted, then p ( ) i (t) =^ti, 0i.The cohomology groups of the cochain complex C 6= (X) =fc (X); g is, in general, uninteresting, as shown in the following theorem [3]. Theorem 1.1. If dim = 0, then H (C 6= (X)) = G (the subgroup of G = Q 2 G, G = G, consisting of those elements having all but a nite number of their -coordinates in G 0 ), and H (C 6= (X)) = 0, when 6= 0. To pass to more interesting cohomology groups, the topology of the space X will be used to dene that ' 2 C (X) is said to be -locally zero on M X if there is 2 X (M) (the set of external covering of M by open subsets of X) such that ' vanishes on \ M, i.e. each ' vanishes on ( \ M) ( )+1, where = [fu : u 2 g. The subgroups of C (X) consisting of those elements which are -locally zero on X, A respectively are denoted by C 0 (X), C (X; A). The -Alexander-Spanier cohomology of (X; A) over (G; G 0 ), denoted by H (X; A), is the cohomology of the uotient cochain complex C 6= (X; A) =C6= (X; A)=C 6= (X). 0 If f : (X; A)! (Y;B) is in Q, 2 (Y;B) and = f 1 ( ), then f denes a cochain map f 6= : C 6= (Y;B)! 6= C (X; A), where (f ')=(') for each ' 2 C (Y ). In turn, f 6= induces the homomorphism f : H (Y;B)! H (X; A). On the other hand, for 2 (X; A), denote by C. The subgroup of C = C () consisting of those ' which vanishes on 0 \ A. Then we obtain a direct system fc 6= g (X;A) such that any map f 2 Q constitutes a map F : fc 6= g (Y;B)!fC 6= g (X;A) [9]; its limit is F 1. Theorem 1.2. The -Alexander-Spanier cohomology functor f H, f g is naturally isomorphic to the functor flim! fh (C 6= )g (X;A), F 1 g [4]. In the previous part, the topology on (G; G) plays no role; to pass to the second cohomology of -type we characterize an element ' 2 C (X) tobe-partially continuous if it is continuous on some 2 (X), i.e. ' j ( )+1 are continuous functions. Let L (X) be the group of all such elements, and M 6= (X) = L6= (X)=C 6= 0 (X). The subgroup of C, where 2 (X), consisting of the -continuous elements ', i.e. ' are continuous, is denoted by M. Let i : A,! X, dene M 6= (X; A) to be the mapping cone of i6= : M 6= (X)! M 6= (A), [13],[18], assuming that M (X; A) =M (X)M 1 (A), and the coboundary is ('; ) =( '; i ' + 1 ). The cohomology of M 6= (X; A) is the partially continuous -Alexander-Spanier cohomology of (X; A) over the topological pair (G; G 0 ) of coecient groups; it is denoted by H ~ (X; A). On the other hand, if 2 ~(X; A), then i denes a cochain map i 6= : M 6=! M 6= 0 ; its mapping cone is denoted by M 6=. 3

4 Theorem 1.3. lim! fm 6= g ~(X;A) [4]. For a pair (X; A) 2 Q with A is closed, M 6= (X; A) is naturally isomorphic to Theorem 1.4 For a discrete space, and 0, ~ H (X) ' H (X). Proof. Since X ( )+1 admits a discrete topology, it follows that each -coordinate ' of ' 2 C (X) is continuous [16]. Then ' is -partially continuous with respect to any 2 (X). Therefore L (X) =C (X) and M 6= (X) = C 6= (X). 2 Tautness and Continuity Properties This article is devoted to study the tautness property for both Alexander-Spanier cohomology of -types. One of its applications is the continuity property. The star of a subset A in a space X with respect to 2 (X) is st(a; ) =[fu 2 : U d \ A 6= ;g The star of is = fst(u ;):u 2g Denition 2.1 Let, 2 (X), then is a star-renement of, written <,if<. Denote by (A) the collections of neighborhoods fg of A in X; it is directed downward by inclusion. If 1 < 2, then the inclusion 12 : 2,! 1 induces the homomorphisms 12 : H ( 1)! H ( 2). Also i : A,! induces i : H ()! H (A), and they dene a homomorphism Theorem 2.1 (Tautness). I 1 : lim! f H (); 12 g (A)! H (A) : A closed subspace of a paracompact space is a taut subspace relative to the -Alexander-Spanier cohomology, i.e. I 1 is an isomorphism for each and any pair (G; G 0 ) of coecient groups. Proof. (1) I 1 is an epimorphism. Actually let h 2 H (A) with representative ' 2 C (A), written h =[']. Let ' 2 C (A) such that ' 2 '. Then there is = fu = \ A : X is openg 2(A) such that Since A is closed, it follows that = f g[fx ( ') j ( )+2 =0 (2:1) Ag 2(X). The paracompactness of X is euivalent to the existence of such 2 (X) that < [21], and a neighborhood of A and an extension f :! A (not necessarily continuous) of the identity map id A of A, i.e. fi =id A,such that f(u \ ) st(u ;) for each u 2 [18]. One can show that f denes a cochain map f 6= : C 6= (A)! C 6= () by (f ') = ' f (( )+1) with (f ') = ('), where 4

5 f ( ) :! A given by f(x 0 ;:::x 1 )=(f(x 0 );:::;f(x 1 )). The relation < yields that for each u 2 there is u 2 such that f(u \ ) st(u ;) u. Because f() =A, then f(u \ ) u \ A u for some u 2. By using (2.1), we get ( f ') j( \ ) ( )+2 =0, i.e. (f ') 2 C +1 0 (). Then f ' represents a cocycle f ' 2 C () which, in turn, denes h 2 H (), i.e. h =[f ']. Let t 2 A ( )+1, then (i (f ')) (t) =' f (()+1) i and therefore i h =[(fi ) ']=[']=h. (( )+1) (t) =' (t); (2) I 1 is a monomorphism. Actually, let h 1 2 H ( 1), ' 1 such that ' 1 2 ' 1, ' 1 2 h 1, and [h 1 ] 2 eri 1. 2 C ( 1) and ' 1 2 C ( 1 ) First, one can consider that the neighborhood 1 of A is a paracompact subset of X. For, if 1 is not so, then there is a paracompact subset M 1 of X such that M 1 < 1 (e.g., take M 1 = X) [10]. The inclusion M 11 induces an epimorphism 6= M11 [3], let M11 1 = ' 1. Thus the cohomology class of H (M 1) represented by 1 is [h 1 ], which shows that 1 can be taken paracompact. ow, ' 1 2 er, or euivalently, there is = fu = \ 1 : X is openg 2( 1 ) such that ( ' 1 ) j ( )+2 =0: (2:2) On the other hand, the assumption i 1 h 1 =0asserts that there exists ' 2 C 1 (A) such that i 1 ' 1 1 ' 2 C 0 (A), where ' 2 '. This means that there is such = fu =! \ A :! X is openg 2(A) that Assume that 1 = fu 1 =! \ 1 g[f 1 (i 1 ' 1) =( 1 ') on ( )+1 (2:3) Ag. The paracompactness of 1 asserts the existence of 1 ; 2 2( 1 ) for which < 1 and 1 < 2. The directedness of ( 1 ) implies that there 2 ( 1 ) for which 1 ; 2 <; and so for each u 2 there are u i 2 i, i =1;2and u 2, u such that u u i st(u i ; i )u \u 1 ; Then st(u ;)u \u 1 (2:4) i.e. ; 1 <. According to Lemma in [18], there is a neighborhood 2 of 1 and f : 2! A (not necessarily continuous) such that fi 2 =id A, and u such that f(u \ 2 ) st(u ;)u 1 u 1 \A=u (2:5) Then, by (2.3), we get ( 1 f 1 ') =(f i 1 ' 1)on(\ 2 ) ()+1 (2:6) 5

6 Dene D : C +1 ( 1 )! C ( 2 )by: if t =(x 0 ;:::;x () )2 ()+1 2 and 1 2 C +1 ( 1 ) then where (D 1) (t) = () X r=0 ( 1) 1 (y 0 ;:::;y ;z ;:::;z () ) ; y j = 12(x j ); z j =(i 1 f)(x j)=f(x j ); and (D 1 )=( 1 ). A similar calculation as given in [4], we get ( 1 D 1 ' 1 ) =(f i 1 ' 1) ( 12 ' 1) (D ' 1 ) (2:7) By (2.4), (2.5) for each u 2, there is u 2 such that (u \ 2 ) [ f(u \ 2 ) u Then, by (2.7), (2.2), (2.6) conseuently, we have ( 1 D 1 ' 1 ) =(f i 1 ' 1) ( 12 ' 1) on ( \ 2 ) ( )+1 ; and so Therefore ( 12 ' 1) =( 1 (f 1 ' D 1 ' 1 )) on ( \ 2 ) ( )+1 : 2 = f 1 ' D 1 ' 1 2 C 1 ( 2 ) such that i.e. 12 h 1 = 0 which completes the proof. ( 12 ' 1) =( 1 2) on ( \ 2 ) ( )+1 ; Corollary 2.2. Any one-point subset of a paracompact is a taut subspace relative to H. The next part is devoted to study the tautness property for ~ H, which is also valid for H. The idea and results of study. -contiguous maps, introduced in [4] plays an essential role in this The inclusions 12 : 2,! 1 corresponding to the relations 1 < 2 in (A), dene the direct system f ~ H (); ~ 12 g. Also the inclusion i : A,!, where 2(A), dene a map of direct systems [9]: I : fh (M 6= ); ~ g ( )!fh (M 6= ~ ; ~ ~ ~ g (A) where 2 (), ~ = i 1 () =\A. On the other hand, f ~ i g dene a homomorphism ~I 1 : lim! f ~ H (); ~ 12 g (A)! ~ H (A) 6

7 Theorem 2.3 (Tautness). If A is a closed subset in a Hausdor space X such that A is a neighborhood retract, then A is a taut subspace relative to the cohomology ~ H. Proof. 1) I ~ 1 is an epimorphism. Actually, let h 2 H ~ (A). Without loss of generality, the neighborhood retractness of A in X yields that A has an open neighborhood U (in X) such that U and a retraction 1 : U! A (If U 1 is an open neighborhood of A of which A is retract but U 1 6, take U = U 1 \ Int). Let i U : A,!U then, I ~ 1 [~ 1(h)] = ~ i U (~ h)= e 1 id A(h) =h. 2) I ~ 1 is a monomorphism. Let [h] 2 er~ I 1, It is sucient to construct V 2(A) satisfying <V and ~ V h =0. Since the cohomology functor commutes with the direct limit [18]. Theorem 1.3 asserts that one may assume that h belongs to lim! fh (M 6= ), ~ g ( ) with representative h 2 H (M 6= ), where = fu =! \ :! X is openg 2() Let 1 = f! g[fx Ag, ~= 1 \A, =fu = 1 1 (u ~)\(u \U):6=u ~ 2~g; V =[u,= 1 jv :V,!A, and 0 = 1 \ V. Then ~ 2 (A), 0 = \ V 2 (V ), u ~ u for each u ~ 6=, is a family of open subsets in Uand so open in X, V is an open neighborhood of A such that V U, and 2 (V ). Since u = u \ u V \ u = u 0, it follows that 0 <. Also 0 \ A = \ A = ~ and j 1 = ~, where j : A,!V. If ` : V,!, and ['] 2 H (M 6= ), then ~ j ~ 0 ~`['] =~ j [f(' j 0 ()+1 )j ( )+1 g] =[f' j~ ()+1 g] ; i.e. ~ j ~ 0 ~` = ~ i ; (2:8) where ~ i 6= ; : M6=! M6= ~ is induced by i : A,!. On the other hand, (j)u u and so j, id V : V! V are contiguous [4]. It follows that ( idv e ) ; ( jr) f : M! M are cochain homotopic [4]. Then ( e idv ) = (f jr) = ~r ~ ~ j, which yields that ~ j is a monomorphism. Because ~ i ; h =0,(2.8) yields that ~ 0 ~` h =0. Since ~` h,~ 0 (~`h ) represent the zero element of lim! fh (M 6= ), ~ g ( ), it follows that ~ V h =[~` h ]=0. The rest of this article is centered around a special case of the continuity property for H. As an application of the continuity property the cohomology groups satisfy a much stronger form of the excision axiom. The following results can be deduced from those given in [9]. 7

8 Lemma 2.4. Let X be the intersection of a nested system fx ; g, then (i) X and limfx ; g are homeomorphic (ii) If the nested system consists of compact Hausdor spaces then X is a closed subset of each X. (iii) If is an open neighborhood of X in X (for some 2 ), then there is >in such that X. The inclusions i : X,! X dene a map its direct limit is denoted by I 1. I : f H (X ); g! H (X) ; Theorem 2.5 (Weak continuity). compact Hausdor spaces, then I 1 is an isomorphism. If X is the intersection of a nested system fx ; g of Proof. Since each X is a paracompact Hausdor space [10] and X is closed in X (Lemma 2.4), it follows, by Theorem 2.1, that X is a taut subspace in X relative to H. (1) I 1 is an epimorphism. Let h 2 H (X), then, according to Theorem 2.1, there exists an open neighborhood of X in X and h 2 H (), such that i (h ) = h. By Lemma 2.4, there is > in such that X. Let i : X,! X, j : X,!. Because i ( j h )=(j i ) h = i h =h, then I 1 [ j h ]=h. (2) I 1 is a monomorphism. Let [h ] 2 er I 1, i.e. i h = 0. The tautness of X in X yields, by Theorem 2.1, an open neighborhood of X in X such that h is the uniue element for which i 0 h =0,where i 0 : X,!. Because i 0 ( i h )= i h =0,then i h =0. Let >in such that X, then h =(i i ) h = j ( i h ) = 0, i.e. [h ]=0. 3 Applications One of the good applications of the Alexander-Spanier cohomology of -types is the study of the 0-dimensional cohomology groups and their relation with the connectedness of the space [4]. In this article two applications are given. In a next work we hope to give more applications. The rst application is concentrated to dene the partially continuous -Alexander-Spanier cohomology of an excision map and calculate its value for some dimensions. Let f ~ 6= : M 6= (Y;B)! M6= (X; A) be the cochain map induced by the map f in Q. Dene M 6= (f) to be the mapping cone of ~ f 6= by: M (f) =M 1 (Y;B) M (X; A) ; = M (Y ) M 1 (B) M 1 (X) M 2 (A) ; and the coboundary is ~ (' 2 ; 2 ;' 1 ; 1 )= 8

9 Then there is a short exact seuence =( (' 2 ; 2 ); (' 1 ; 1 )+ ~ f (' 2 ; 2 )) =( ' 2 ; ~ i ' 2 1 2; 1 ' 1 + ~ f ' 2 ; ~ i 1 ' g fja) 1 2) 0! + M 6= (X; A) 6=! M 6= (f) x6=! M 6= (Y;B)! O 2 (3:1) where 6=, 6= + are injection, projection respectively; M 6= (X; A) is the complex M 6= (X; A) with the dimensions all raised by one, and M 6= (Y;B) is the complex M 6= (Y;B) with the sign of the coboundary changed [12]. ote that H ( M 6= (Y;B)) = ~ H (Y;B). Let V be an open subset of X such that V IntA; B = X V, and C = A V. Put the excision map e :(B;C),! (X; A) in (3.1) instead of f, and then apply the cohomology functor, we get the long exact seuence: :::! ~ H (e) ~! ~ H (X; A) ~e! ~ H (B;C) ~! H ~ +1 (e)! ::: (3:2) Thus the groups H ~ (e), ~ +1 H (e) measure how much the cohomological groups deviate from the excision axiom. Theorem 3.1. If dim =0,e:(B;C),! (X; A) is an excision map, where A is closed and (G; G 0 )any pair of topological abelian groups, then H ~ (e) = 0 when =0or=1. Proof. (1) Case =0. Wehave M(e)=M 0 (X; 0 A) =M(X)=L 0 0 (X) Let ' 2 M 0 (e) such that ~' = 0, then ~ i 0 ' =0,~e' =0. Then ' = 0 [4], which means that er ~ 0 =0. (2) Case =1. Wehave M(e)=M 1 (X)L 0 0 (A)L 0 (B): It is sucient to show that er ~ 1 Im ~ 0. Let (' 2 ; 2 ;' 1 ;0) 2 er~ 1, then 1 ' =0; ~ i 0 ' 2 = 0 2 ~e 1 ' 2 = 0 ' 1 (3:3) ~e 0 1( 2 )=~ j'1 (3:4) where i : A,!X,j:C,!Band e 1 = ejc. By (3.4), there exists [4], ' 2 M 0 (X) =L0 (X) such that ~ i 0 ' = 2 ; ~e 0 ' = ' 1 (3:5) 9

10 By (3.3)-(3.5), we get ~ i 1 ( 0 ' ' 2 )=0; ~e 1 ( 0 ' ' 2 )=0 (3:6) Then 0 ' = ' 2 [4], which with (3.6) yield that ('; 0; 0; 0) 2 M 0 (e) such that ~ 0 ('; 0; 0; 0) = (' 2 ; 2 ;' 1 ;0). Combining the seuence (3.2) and the above theorem, we get the following result. Corollary 3.2 Under the assumptions of Theorem (3.1), the map ~e 0 : ~ H 0 (X; A)! ~ H 0 (B;C) is an isomorphism but ~e 1 is a monomorphism: The second application is to give attention in our work to use a pair of coecients groups, an arbitrary locally-nite simplicial complex, and the condition (k). Let : (G; G 0 )! (F; F 0 ) be a homeomorphism of pairs of (discrete) abelian groups which is an epimorphism, (L; L 0 ) = er and :(L; L 0 ),! (G; G 0 ). Then for each 2 (X; A), the maps, dene, naturally a short exact seuence 0! C (; L; L 0 )! C (;G; G 0 )! C (;F; F 0 )! 0; its cohomology is a long exact seuence [12] denoted by S. One can show that fs g (X;A) is a direct system, its direct limit [3], [4] :::! H 1 (X; A; F; F 0 )! H (X; A0 ; L; L 0 )! H (X; A; G; G0 )! H (X; A; F; F 0 )! H +1 (X; A; L; L0 )! ::: ow instead of F take the factor group G=G 0 and so instead of F 0 will be the null subgroup of G=G 0. Then the above seuence yields the following result. Theorem 3.3 Consider that (X; A) has a trivial ( 1)-dimensional -Alexander-Spanier cohomology group with nite cochains, and a trivial ( + 1)-dimensional -Alexander-Spanier cohomology with innite cochains, taken over the coecient groups G=G 0 and G 0 respectively. Then the group H (X; A; G; G0 ) dened over an arbitrary pair (G; G 0 ) of coecient groups is the extension of the cohomology group H (X; A; G0 ) with innite cochains over G 0 by the group H (X; A; G=G0 ) with nite cochains over G=G 0. Acknowledgments One of the authors (A.El-S.D.) would like to thank the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy for hospitality. This work was done within the framework of the Associateship Scheme of the Abdus Salam ICTP. 10

11 References [1] Abd El-Sattar A. Dabbour, \On the Baladze homology groups of compact spaces", Soobsc. Akad. auk Gruzin. SSR 77, o.3 (1975), [2] Abd El-Sattar A. Dabbour, \On the -olmogorov homology and cohomology construction", Simon Stevin Math. J. 66, o.3-4 (1992), [3] Abd El-Sattar A. Dabbour and Rola A. Hijazi, \The -Alexander-Spanier cohomology", ICTP preprint IC/95/231; Bull. Pure and Appl. Sci. 17 (1998). [4] Abd El-Sattar A. Dabbour and Rola A. Hijazi, \Axiomatic characterization of Alexander- Spanier -types"" ICTP preprint IC/96/152; Bull. Pure and Appl. Sci. 17 (1998). [5]. Berikashvili, \On axiomatic theory of spectra and duality laws for arbitrary space", Trudy Tbilisi Mat. Inst. 24 (1957). [6] E.H. Brown Jr. and R.H. Szczarba, \Continuous cohomology and real homotopy type", Trans. Amer. Math. Soc. 311, o. 1 (1989) [7] E.H. Brown Jr. and R.H. Szczarba, \Continuous cohomology II", Asterisue 191 (1990). [8] G.S. Chogoshvili, \On homological approximation and duality laws for arbitrary space", Mat. Sbornik 28 (1951). [9] S. Eilenberg and. Steenrod, Foundation of Algebraic Topology, Princeton Univ. Press (1952). [10] R. Engelking, General Topology, PW Warsaw (1977). [11] S.T. Hu, \Cohomology theories on topological groups", Mich. Math. J. 1 (1952), [12] S. MacLane, Homology, Springer-Verlag (1994). [13] W.S. Massey, A Basic Course on Algebraic Topology, Springer-Verlag (1991). [14] L.M. Mdzinarishvili, \Partially continuous Alexander-Spanier cohomology theory", Univ. Heidelberg, Math. Inst, Vol. 130 (1996), [15] M.A. Mostow, \Continuous cohomology of spaces with two topologies", Mem. Amer. Math. Soc. 175 (1976). [16] C.W. Patty, Foundation of Topology", PWS-ent Publ. Comp. (1993). [17] E.H. Spanier, \Cohomology theory for general spaces", Ann. Math. 49, o.2 (1948), [18] E.H. Spanier, Algebraic Topology, Springer-Verlag (1966). 11