Summer Conference on Topology and Its Applications
 

Document Type

Topology + Foundations

Publication Date

6-2017

Publication Source

32nd Summer Conference on Topology and Its Applications

Abstract

In the paper the necessary and sufficient conditions are found under which a metrizable space has the Stone-Cech compactification whose remainder has the given cohomological dimensions (cf. [Sm], Problem I, p.332 and Problem II, p.334, and [A-N]).

In the paper [B] an outline of a generalization of Cech homology theory was given by replacing the set of all finite open coverings in the definition of Cech (co)homology group (Ĥnf(X, A;G)) Ĥnf(X, A;G) (see [E-S], Ch.IX, p.237) by the set of all finite open families of border open coverings [Sm1].

Following Y. Kodama (see the appendix of [N]), we give the following definition:

Definition 1. The border small cohomological dimension d∞f(X;G) of normal space X with respect to group G is defined to be the smallest integer n such that, whenever m ≥ n and A is closed in X, the homomorphism i*A, ∞:Ĥ∞m(X;G)→ [^(H)]∞m(A;G) induced by the inclusion i:A→ X is an epimorphism.

The border small cohomological dimension of X with coefficient group G is a function df∞:N→ N∪{0, + ∞}:X→ n, where d∞f(X;G)=n and N is the set of all positive integers.

We have the following results:

Theorem 2. Let X be a metrizable space. Then the following equality df∞(X;G) = df(βX\X;G)

holds, where df(βX\X;G) is the small cohomological dimension of βX\X (see [N], p.199).

Theorem 3. Let A be a closed subspace of a normal space X. Then df∞(A;G) ≤ df∞(X;G).

Corollary 4. For each closed subspace A of a metrizable space X, df∞(A;G) ≤ df(βX\X;G).

Definition 5. The border large cohomological dimension D∞f(X;G) of normal space X with respect to group G is defined to be the largest integer n such that Ĥ∞n(X, A;G) ≠ 0 for some closed set A of X.

The border large cohomological dimension of X with coefficient group G is a function D∞f:N→ N∪{0, + ∞}:X→ n, where D∞f(X;G)=n and N is the set of all positive integers.

Theorem 6. For each metrizable space X, one has D∞f(X;G) = Df(βX\X;G),

where Df(βX\X;G) is the large cohomological dimension of βX\X (see [N], p.199).

Theorem 7. If A is a closed subset of normal space X, then D∞f(A;G) ≤ D∞f(X;G).

Corollary 8. For each closed subspace A of metrizable space X, one has Df∞(A;G) ≤ Df(βX\X;G).

Theorem 9. If X is a normal space, then d∞f(X;G) ≤ D∞f(X;G).

Corollary 10. For each metrizable space X, one has df(βX\X;G) ≤ Df∞(X;G)

and df∞(X;G) ≤ Df(βX\X;G).

Remark 11.The results of this paper also hold for spaces satisfying the compact axiom of countability [Sm1]. The locally metrizable spaces, complete in the seance of Cech spaces and locally compact spaces satisfy the compact axiom of countability.

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