\v{C}ech border homology and cohomology groups and some applications
Vladimer Baladze

TL;DR
This paper develops cech border homology and cohomology groups for closed pairs of normal spaces, providing intrinsic characterizations related to open coverings, cyclicity, and dimensions of Stone-cech remainders.
Contribution
It introduces new cech border homology and cohomology groups and explores their applications in characterizing various topological properties.
Findings
Characterization of cech homology and cohomology via border groups
Connections between border groups and dimensions of Stone-cech remainders
Applications to normal spaces and their compactifications
Abstract
In the paper the \v{C}ech border homology and cohomology groups of closed pairs of normal spaces are constructed and investigated. These groups give intrinsic characterizations of \v{C}ech homology and cohomology groups based on finite open coverings, homological and cohomological coefficients of cyclicity, small and large cohomological dimensions of remainders of Stone-\v{C}ech compactifications of metrizable spaces.
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Taxonomy
TopicsTopological and Geometric Data Analysis · Homotopy and Cohomology in Algebraic Topology
Čech border homology and cohomology groups and some applications††The author was supported by grant FR/233/5-103/14 from Shota Rustaveli National Science Foundation (SRNSF)
Vladimer Baladze
Department of Mathematics
Batumi Shota Rustaveli State University
Abstract
In the paper the Čech border homology and cohomology groups of closed pairs of normal spaces are constructed and investigated. These groups give intrinsic characterizations of Čech homology and cohomology groups based on finite open coverings, homological and cohomological coefficients of cyclicity, small and large cohomological dimensions of remainders of Stone-Čech compactifications of metrizable spaces.
Keywords and Phrases: Čech homology, Čech cohomology, Stone-Čech compactification, remainder, cohomological dimension, coefficient of cyclicity.
Introduction
The investigation and discussion presented in this paper are centered around the following problem:
Find necessary and sufficient conditions under which a space of given class has a compactification whose remainder has the given topological property (cf. [Sm2], Problem I, p.332 and Problem II, p.334).
This problem for different topological invariants and properties was studied by several authors:
J.M.Aarts [A], J.M.Aarts and T.Nishiura [A-N], Y. Akaike, N. Chinen and K. Tomoyasu [Ak-Chin-T], V.Baladze [B1], M.G. Charalambous [Ch], A.Chigogidze ([Chi1], [Chi2]), H. Freudenthal ([F1],[F2]), K.Morita [Mo], E.G. Skljarenko [Sk], Ju.M.Smirnov ([Sm1]-[Sm5]) and H.De Vries [V] found conditions under which the spaces have extensions whose remainders have given covering and inductive dimensions, and combinatorial properties.
The remainders of finite order extensions are defined and investigated by H.Inasaridze ([I1], [I2]). Using the results obtained in these papers, H.Inasaridze [I3], L.Zambakhidze ([Z1],[Z2]), and I.Tsereteli [Ts] solved interesting problems of homological algebra, general topology and dimension theory.
-dimensional (co)homology groups of remainders of precompact spaces are studied by V.Baladze [B3], V.Baladze and L.Turmanidze [B-Tu].
A.Calder [C] described -dimensional cohomotopy groups of remainders of Stone-Čech compactifications.
The characterizations of shapes of remainders of spaces are established in papers of V.Baladze ([B2],[B3]), B.J.Ball [Ba], J.Keesling ([K1], [K2]), J.Keesling and R.B. Sher [K-Sh].
The present paper is motivated by the general problem mentioned above. Specifically, we study this problem for the properties: Čech (co)homology groups based on finite open covers, coefficients of cyclicity and cohomological dimensions of remainders of Stone-Čech compactifications of metrizable spaces are given groups and given numbers, respectively.
In this paper we define the Čech type covariant and contravariant functors which coefficients in an abelian group ,
[TABLE]
and
[TABLE]
from the category of closed pairs of normal spaces and proper maps to the category of abelian groups and homomorphisms. The construction of these functors is based on all border open covers of pairs (see Definition 1.1 and Definition 1.2).
One of our main results of the paper is the following theorem (see Theorem 2.1). Let be the category of closed pairs of metrizable spaces and proper maps. For each closed pair , one has
[TABLE]
and
[TABLE]
where and are Čech homology and cohomology groups based on all finite open covers of , respectively (see [E-St], Ch. IX, p.237).
We also consider the border cohomological and homological coefficients of cyclicity and , border small and large cohomological dimensions and and prove the following relations (see Theorem 2.3, Theorem 2.5 and Theorem 2.8):
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where , and , are well known cohomological coefficient of cyclicity [No], homological coefficient of cyclicity (see Definition 2.2) and small cohomological dimension, large cohomological dimension [N] of remainders and , respectively.
Without any further reference we will use definitions and results from the monographs General Topology [En], Algebraic Topology [E-St] and Dimension Theory [N].
1 On Čech border homology and cohomology groups
In this section we give an outline of a generalization of Čech homology theory by replacing the set of all finite open coverings in the definition of Čech (co)homology group () (see [E-St],Ch.IX, p.237) by a set of all finite open families with compact enclosures. For this aim we give the following definitions.
An indexed family of subsets of set is a function from an indexed set to the set of subsets of . The image of index is denoted by . Thus the indexed family is the family . If , then we say that family is a finite family.
Let be a subset of set . A family is called a subfamily of family .
By we denote the family consisting of family and its subfamily .
Definition 1.1**.**
(see [Sm4]). A finite family of open subsets of normal space is called a border cover of if its enclosure is a compact subset of .
Definition 1.2**.**
(cf. [Sm4]). A finite open family is called a border cover of closed pair if there exists a compact subset of such that and .
The set of all border covers of is denoted by . Let . Then the family is a border cover of subspace .
Definition 1.3**.**
Let be two border covers of with indexing pairs and , respectively. We say that the border cover is a refinement of border cover if there exists a refinement projection function such that for each index () .
It is clear that becomes a directed set with the relation whenever is a refinement of .
Note that for each , , and if for each , and , then .
Let be two border covers with indexing pairs and , respectively. Consider a family , where and . Let , where , . Assume that . The family is a border cover of and .
For each border cover with indexing pair , by denote the nerve , where is the subcomplex of simplexes of complex with vertices of such that , where is the carrier of simplex (see [E-St], pp.234). The pair is a simplicial pair. Moreover, any two refinement projection functions induce contiguous simplicial maps of simplicial pairs (see [E-St], pp. 234-235).
Using the construction of formal homology theory of simplicial complexes ([E-St], Ch.VI) we can define the unique homomorphisms
[TABLE]
and
[TABLE]
where is any abelian coefficient group.
Note that and . If than
[TABLE]
and
[TABLE]
Thus, the families
[TABLE]
and
[TABLE]
form inverse and direct systems of groups.
The inverse and direct limit groups of above defined inverse and direct systems are denoted by symbols
[TABLE]
and
[TABLE]
and called -dimensional Čech border homology group and -dimensional Čech border cohomology group of pair with coefficients in abelian group , respectively.
According to [E-St] a border cover indexed by is called proper if is the set of all with . The set of proper border covers is denoted by . Now define a function
[TABLE]
By definition, for each border cover of
[TABLE]
where is the set of for which . It is clear that the family is a proper border cover and the function induced by is one to one. Moreover, if , then .
Proposition 1.4**.**
For each pair the set of proper border covers of is a cofinal subset of .
Proof.
Let be a border cover of . Assume that
[TABLE]
Consider a family consisting of subsets
[TABLE]
and
[TABLE]
Note that is a border cover of and . ∎
Consequently, in definitions of Čech border homology and cohomology groups of pairs we may replace the set by the subset .
Now we define, for a given proper map of pairs, the induced homomorphisms
[TABLE]
and
[TABLE]
Let be a border cover with index set and . Consider a family . Note that
[TABLE]
Let and . Since is proper, is a compact subset of .
Since , the subfamily is such that . Let and . Note that Hence, is a border cover of pair .
It is clear that is a subcomplex of and is a subcomplex of . By a symbol denote the simplicial inclusion of into .
If and , then the diagrams
H_{n}(X_{\beta^{{}^{\prime}}},A_{\beta^{{}^{\prime}}};G)$$H_{n}(X_{\beta},A_{\beta};G)$$H_{n}(X_{\alpha^{{}^{\prime}}},A_{\alpha^{{}^{\prime}}};G)$$H_{n}(X_{\alpha},A_{\alpha};G)$$f_{\beta*}$$p_{\alpha^{{}^{\prime}}*}^{\beta^{{}^{\prime}}}$$f_{\alpha*}$$p_{\alpha*}^{\beta}
and
H^{n}(X_{\alpha},A_{\alpha};G)$$H^{n}(X_{\alpha^{{}^{\prime}}},A_{\alpha^{{}^{\prime}}};G)$$H^{n}(X_{\beta},A_{\beta};G)$$H^{n}(X_{\beta^{{}^{\prime}}},A_{\beta^{{}^{\prime}}};G).$$f_{\alpha}^{*}$$p_{\alpha}^{\beta*}$$f_{\beta}^{*}$$p_{\alpha^{{}^{\prime}}}^{\beta^{{}^{\prime}}*}
commute.
Thus, for each , the induced homomorphisms and together with function given by formula
[TABLE]
form maps
[TABLE]
and
[TABLE]
The limits of maps and are denoted by
[TABLE]
and
[TABLE]
and called homomorphisms induced by proper map .
Note that if is the identity map, then the induced homomorphisms and are the identity homomorphisms. Furthermore, for each proper maps and
[TABLE]
and
[TABLE]
We have the following theorem.
Theorem 1.5**.**
There exist the covariant and contravariant functors
[TABLE]
and
[TABLE]
given by formulas
[TABLE]
[TABLE]
and
[TABLE]
[TABLE]
Proof.
The proof follows from above discussion. ∎
We will call the functors and Čech border homology and cohomology functors, respectively.
Now we define boundary and coboundary homomorphisms
[TABLE]
and
[TABLE]
Let , and . The refinement projection functions induce the unique homomorphisms and , and , which form inverse systems
and
and direct systems
and
Let
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Our main aim is to show that the groups and , and , and , and are isomorphical groups.
Next we define a function . Let . Assume that for . We have defined the border cover indexed by pair .
Let . Note that
[TABLE]
It is clear that is a compact subset of the subspace . Thus, . The defined function is an order preserving function.
It is easy to show that the image of function is a cofinal subset of set . Note that . By denote this simplicial isomorphism. Hence, the family of pairs induces a map of inverse systems and direct systems
[TABLE]
and
[TABLE]
Let and . Since all homomorphisms and are isomorphisms, the limit homomorphisms
[TABLE]
and
[TABLE]
are isomorphisms.
Lets us also define a function . For each assume that , . The family is indexed by and .
Note that . Let be a simplicial isomorphism. The family of pairs induce the maps of inverse and direct systems
[TABLE]
and
[TABLE]
Let and . Since each and are isomorphisms, the induced limit homomorphisms
[TABLE]
and
[TABLE]
are isomorphisms.
There exist the limit sequences
\cdots$$\check{H}_{n}^{\infty}(X,A;G)$$\check{H}_{n}^{\infty}(X;G)_{(X,A)}$$\check{H}_{n}^{\infty}(A;G)_{(X,A)}$$\check{H}_{n+1}^{\infty}(X,A;G)$$\cdots$$j_{n}^{{}^{\prime}\infty}$$i_{n}^{{}^{\prime}\infty}$$\partial^{{}^{\prime}\infty}_{n+1}
and
\cdots$$\hat{{H}}^{n}_{\infty}(X,A;G)$$\hat{{H}}^{n}_{\infty}(X;G)^{(X,A)}$$\hat{{H}}^{n}_{\infty}(A;G)^{(X,A)}$$\hat{{H}}^{n+1}_{\infty}(X,A;G)$$\cdots$$j^{{}^{\prime}n}_{\infty}$$i^{{}^{\prime}n}_{\infty}$$\delta_{\infty}^{{}^{\prime}n}
generated by the families consisting of homology and cohomology sequences of simplicial pairs , respectively.
Consider the diagrams
\check{H}_{n}^{\infty}(X,A;G)$$\check{H}_{n-1}^{\infty}(A;G)_{(X,A)}$$\check{H}_{n-1}^{\infty}(A;G)$$\partial^{{}^{\prime}\infty}_{n}$$\Phi_{n-1}
and
\hat{H}^{n}_{\infty}(A;G)$$\hat{H}^{n}_{\infty}(A;G)^{(X,A)}$$\hat{H}_{\infty}^{n+1}(X,A;G)$$\Psi^{n}$$\delta^{{}^{\prime}n}_{\infty}
and define the boundary homomorphism of Čech border homology groups and coboundary homomorphism of Čech border cohomology groups as compositions
[TABLE]
and
[TABLE]
In this way we arrive to the following theorems.
Theorem 1.6**.**
Let be a proper map. Then hold the following equalities
[TABLE]
and
[TABLE]
Proof.
The desired equalities follow from the commutativity of the diagrams.
\check{H}_{n}^{\infty}(X,A;G)$$\check{H}_{n-1}^{\infty}(A;G)_{(X,A)}$$\check{H}_{n-1}^{\infty}(A;G)$$\check{H}_{n}^{\infty}(Y,B;G)$$\check{H}_{n-1}^{\infty}(B;G)_{(Y,B)}$$\check{H}_{n-1}^{\infty}(B;G)$$\partial_{n}^{{}^{\prime}\infty}$$\Phi_{n-1}$$\partial_{n}^{{}^{\prime}\infty}$$\Phi_{n-1}$$f_{*}^{\infty}$$(f_{|A})_{*}^{{}^{\prime}\infty}$$(f_{|A})_{*}^{\infty}
and
\hat{H}^{n-1}_{\infty}(B;G)$$\hat{H}^{n-1}_{\infty}(B;G)^{(Y,B)}$$\hat{H}^{n}_{\infty}(Y,B;G)$$\hat{H}^{n-1}_{\infty}(A;G)$$\hat{H}^{n-1}_{\infty}(A;G)^{(X,A)}$$\hat{H}^{n}_{\infty}(X,A;G),$$\Phi^{n-1}$$\delta^{{}^{\prime}n}_{\infty}$$\Phi^{n-1}$$\delta^{{}^{\prime}n}_{\infty}$$(f_{|A})^{*}_{\infty}$$(f_{|A})^{{}^{\prime}*}_{\infty}$$f^{*}_{\infty}
where and are defined as the appropriate limit homomorphisms. ∎
Let and be the inclusion maps.
Theorem 1.7**.**
Let . Then the Čech border cohomology sequence
\cdots$$\check{H}^{n-1}_{\infty}(A;G)$$\check{H}^{n}_{\infty}(X,A;G)$$\check{H}^{n}_{\infty}(X;G)$$\check{H}^{n}_{\infty}(A;G)$$\cdots$$\delta_{\infty}^{n-1}$$j^{*}_{\infty}$$i^{*}_{\infty}
is exact while the Čech border homology sequence
\cdots$$\hat{H}_{n-1}^{\infty}(A;G)$$\hat{H}_{n}^{\infty}(X,A;G)$$\hat{H}_{n}^{\infty}(X;G)$$\hat{H}_{n}^{\infty}(A;G)$$\cdots$$\partial^{\infty}_{n}$$j_{*}^{\infty}$$i_{*}^{\infty}
is partially exact.
Proof.
One can prove this theorem analogously to the corresponding theorem of the classical Čech theory [E-St]. ∎
Theorem 1.8**.**
Let and be an abelian group. If is open in and , then the inclusion map induces isomorphisms
[TABLE]
and
[TABLE]
Proof.
Let be the subset of consisting of all covers with property:
if , then and .
First we prove that is cofinal in . Let be a border cover of with enclosure . Let be a set such that and there exists a bijective function between and . Let . The correspondence element of in denote by . Define the border cover . Let
[TABLE]
and
[TABLE]
It is clear that is a border cover of with enclosure and .
We prove that is cofinal in . Let be a border cover of with enclosure . Define a border cover .
Let
[TABLE]
The family is a border cover of with enclosure .
Let be a border cover such that . It is clear that .
Note that
[TABLE]
and
[TABLE]
As in [E-St] we can prove that there exist isomorphisms
[TABLE]
and
[TABLE]
The conclusion of the theorem is a consequence of these isomorphisms. ∎
Theorem 1.9**.**
If is a compact space, then for each ,
[TABLE]
and
[TABLE]
Proof.
Let be the border cover of consisting of empty set. It is clear that is a refinement of any border cover of . The set is a cofinal subset of . Consider the inverse system and direct system . We have
[TABLE]
and
[TABLE]
The nerve consists of one vertex. Using the methods of proofs of results VI.3.8 and VI.4.3 of [E-St] we than conclude that
[TABLE]
and
[TABLE]
∎
Thus, Čech border homology (cohomology) functors satisfy the Steenrod-Eilenberg type axioms (cf.[E-St]): axiom of natural transformation, axiom of partially exactness (axiom of exactness), axiom of excision and axiom of dimension; but they do not satisfy the proper homotopy axiom.
The above obtained results yield the next theorem.
Theorem 1.10**.**
Let be a triple of normal space and its closed subsets and with . Then the Čech border homology sequence
\cdots$$\check{H}_{n-1}^{\infty}(A,B;G)$$\check{H}_{n}^{\infty}(X,A;G)$$\check{H}_{n}^{\infty}(X,B;G)$$\check{H}_{n}^{\infty}(A,B;G)$$\cdots$$\bar{\partial}^{\infty}_{n}$$\bar{j}_{*}^{\infty}$$\bar{i}_{*}^{\infty}
and the Čech border cohomology sequence
\cdots$$\hat{H}^{n-1}_{\infty}(A,B;G)$$\hat{H}^{n}_{\infty}(X,A;G)$$\hat{H}^{n}_{\infty}(X,B;G)$$\hat{H}^{n}_{\infty}(A,B;G)$$\cdots$$\bar{\delta}_{\infty}^{n}$$\bar{j}^{*}_{\infty}$$\bar{i}^{*}_{\infty}
are partially exact and exact, respectively. Here , and and are the homomorphisms induced by the inclusion maps , and .
Proof.
The proof is similar to the proof of the corresponding Theorems 10.2 and 10.2c of [E-St] (see Ch. I,§† 10). ∎
2 On some applications of Čech border homology and cohomology groups
Now we are mainly interested in the following problem: how to characterize the Čech homology and cohomology groups, coefficients of cyclicity, and cohomological dimensions of remainders of Stone-Čech compactifications of spaces.
Our main result about the connection between Čech (co)homology groups of remainders and Čech border (co)homology groups of spaces is.
Theorem 2.1**.**
Let and let be the pair of Stone-Čech compactifications of and . Then
[TABLE]
and
[TABLE]
Proof.
Let and be the closed covers of pairs and . By Lemma 4 of [Sm4] there exist open swellings and of and in , respectively. Assume that , . Let
[TABLE]
Note that for each . It is clear that is a swelling of and .
The swelling in of closed cover of is denoted by . Let be the set of all swellings of such kind.
Now define an order in . By definition,
[TABLE]
It is clear that is directed by . Let be the nerve of and be the projection simplicial map induced by the refinement . Consider an inverse system
[TABLE]
and a direct system
[TABLE]
Let be the function in the set of closed finite covers of pair given by formula
[TABLE]
Note that is an increasing function and
[TABLE]
For each index , we have
[TABLE]
and
[TABLE]
It is known that for normal spaces the Čech (co)homology groups based on finite open covers and on finite closed covers are isomorphic. By Theorems 3.14 and 4.13 of ([E-St],Ch.VIII) we have
[TABLE]
and
[TABLE]
For each swelling , the family
[TABLE]
is a border cover of .
Let be the function defined by formula
[TABLE]
The function increases and is a cofinal subset of . Note that the correspondence
[TABLE]
induces an isomorphism of pairs of simplicial complexes. Thus, for each , we have the isomorphisms
[TABLE]
and
[TABLE]
By Theorems 3.15 and 4.13 of ([E-St],Ch.VIII), we have
[TABLE]
and
[TABLE]
From (1), (2), (3) and (4) it follows that
[TABLE]
and
[TABLE]
∎
The cohomological coefficient of cyclicity of pair was defined by S.Novak [N] and M.F.Bokstein [Bo]. Dually one can define the homological coefficient of cyclicity of pair .
Now give the following definitions and results.
Definition 2.2**.**
Let be an abelian group and nonnegative integer. A border (co)homological coefficient of cyclicity of pair with respect to denoted by is , if for all and .
Finally, if for every there is with .
Theorem 2.3**.**
For each pair ,
[TABLE]
and
[TABLE]
Proof.
This is an immediate consequence of Theorem 2.1. Indeed, let . Then for each , and . From the isomorphism
[TABLE]
it follows that for each , and . Thus, .
Analogously we can prove equality ∎
The theory of cohomological dimension has become an important branch of dimension theory since A. Dranishnikov solved P.S. Alexandrov’s problem and developed the theory of extension dimension ([D], [D-Dy]).
Our next aim is to study some questions of theory of cohomological dimension. In particular, we now give a description of cohomological dimension of remainder of Stone-Čech compactification of metrizable space.
Following Y. Kodama (see the appendix of [N]) and T. Miyata [Mi] we give the following definition.
Definition 2.4**.**
The border small cohomological dimension of normal space with respect to group is defined to be the smallest integer such that, whenever and is closed in , the homomorphism induced by the inclusion is an epimorphism.
The border small cohomological dimension of with coefficient group is a function , where and is the set of all positive integers.
Theorem 2.5**.**
Let be a metrizable space. Then the following equality
[TABLE]
holds, where is the small cohomological dimension of (see ).
Proof.
Let be a closed subset of . Assume that . Then for each the homomorphism is an epimorphim. Consider the following commutative diagram
\hat{\rm{H}}^{m}_{\infty}(X;G)$$\hat{\rm{H}}^{m}_{f}(\beta X\setminus X;G)$$(5)$$\hat{\rm{H}}^{m}_{\infty}(A;G)$$\hat{\rm{H}}^{m}_{f}(\beta A\setminus A;G).$$i_{A,\infty}^{*}$$i_{\beta A\setminus A}^{*}$$\approx$$\approx
It is clear that the homomorphim
[TABLE]
also is an epimorphim for each . Thus,
[TABLE]
Let . To see the reverse inequality, let be a closed subset of and let .
Consider an open in neighbourhood of . There exists an open neighbourhood of in such that . By Lemma 5 of [Sm4] we can find an open set in such that and . Let . It is clear that .
We have
[TABLE]
Consequently, . This shows that
[TABLE]
Hence, we have
[TABLE]
Thus, for each closed set of and its open neighbourhood in there exists a closed subset in such that .
Let . There is a closed finite cover of such that an element represents the element .
Using Lemma 4 of [Sm4] we can find the swellings and of in and , respectively, such that . Let be the union of elements of . There is a closed set of with . The nerves , and are isomorphic. We can assume that
[TABLE]
Hence, the element also belongs to the group . Consequently, it represents some element of .
The inclusion induces an epimorphism . From diagram it follows that the homomorphism is an epimorphism. Consequently, there is an element such that . The homomorphism induced by the inclusion satisfies the condition . From equality it follows that .
Thus the inclusion also induces an epimorphism . Hence, we obtaine
[TABLE]
From the inequalities and it follows that
[TABLE]
∎
Theorem 2.6**.**
Let be a closed subspace of a normal space . Then
[TABLE]
Proof.
Let be an arbitrary closed subset of and , and be the inclusion maps. Note that . The induced homomorphisms , and satisfy the equality .
Let . For each , the homomorphisms and are epimorphisms. Hence, the homomorphism is also an ephimorphism for each . Thus, . ∎
Corollary 2.7**.**
For each closed subspace of a metrizable space ,
[TABLE]
Definition 2.8**.**
The border large cohomological dimension of normal space with respect to group is defined to be the largest integer such that for some closed set of .
The border large cohomological dimension of with coefficient group is a function , where and is the set of all positive integers.
Theorem 2.9**.**
For each metrizable space , one has
[TABLE]
where is the large cohomological dimension of (see ).
Proof.
Let . Consider an arbitrary closed subspace of . The remainder is a closed subset of . By the assumption, we have for each . Theorem 2.1 implies that for each and . Thus,
[TABLE]
Let . Assume that . Then there is a closed set in such that . Using Lemma 4 of [Sm4] and the proof of Theorem 2.5 we can show that there is a closed set of such that , and . By Theorem 2.1 . But it is not possible because . Therefore, . Thus,
[TABLE]
The inequalities (8) and (9) imply
[TABLE]
∎
Theorem 2.10**.**
If is a closed subset of normal space , then
[TABLE]
Proof.
By Theorem 1.10, for each closed set of , there is the exact Čech border cohomological sequence
\cdots$$\hat{H}^{m-1}_{\infty}(A;G)$$\hat{H}^{m}_{\infty}(X,A;G)$$\hat{H}^{m}_{\infty}(X,B;G)$$\hat{H}^{m}_{\infty}(A,B;G)$$\cdots$$\bar{\delta}_{\infty}^{m}$$\bar{j}^{*}_{\infty}$$\bar{i}^{*}_{\infty}
It is clear that, if , then . Consequently, . Thus, we have
[TABLE]
∎
Corollary 2.11**.**
For each closed subspace of metrizable space , one has
[TABLE]
Theorem 2.12**.**
If is a normal space then
[TABLE]
Proof.
Let be a closed subset of normal space . Consider the exact Čech border cohomological sequence of pair
\cdots$$\hat{H}^{m-1}_{\infty}(A,B;G)$$\hat{H}^{m}_{\infty}(X,A;G)$$\hat{H}^{m}_{\infty}(X;G)$$\hat{H}^{m}_{\infty}(A;G)$$\cdots$${\delta}_{\infty}^{m}$${j}^{*}_{\infty}$${i}^{*}_{\infty}
Let . Note that is an epimorphism. Hence,
[TABLE]
∎
Corollary 2.13**.**
For each metrizable space , one has
[TABLE]
and
[TABLE]
Remark 2.14**.**
The results of this paper also hold for spaces satisfying the compact axiom of countability. Recall that a space satisfies the compact axiom of countability if for each compact subset there exists a compact subset such that and has a countable or finite fundamental systems of neighbourhoods (see Definition 4 of [Sm4], p.143). A space is complete in the sense of Čech if and only if it is type set in some compact extension. Each locally metrizable spaces, complete in the seance of Čech spaces [Č] and locally compact spaces satisfy the compact axiom of countability.
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