#### 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 (Ĥ^{n}_{f}(X, A;G)) Ĥ_{n}^{f}(X, A;G) (see [E-S], Ch.IX, p.237) by the set of all finite open families of border open coverings [Sm_{1}].

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 d^{f}_{∞}:*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 d^{f}_{∞}(X;G) = d_{f}(βX\X;G)

holds, where d_{f}(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 d^{f}_{∞}(A;G) ≤ d^{f}_{∞}(X;G).

**Corollary 4**. For each closed subspace A of a metrizable space X, d^{f}_{∞}(A;G) ≤ d_{f}(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) = D_{f}(βX\X;G),

where D_{f}(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 D^{f}_{∞}(A;G) ≤ D_{f}(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 d_{f}(βX\X;G) ≤ D^{f}_{∞}(X;G)

and d^{f}_{∞}(X;G) ≤ D_{f}(βX\X;G).

**Remark 11**.The results of this paper also hold for spaces satisfying the compact axiom of countability [Sm_{1}]. The locally metrizable spaces, complete in the seance of Cech spaces and locally compact spaces satisfy the compact axiom of countability.

#### Copyright

Copyright © 2017, the Authors

#### eCommons Citation

Baladze, Vladimer, "On Cohomological Dimensions of Remainders of Stone-Čech Compactifications" (2017). *Summer Conference on Topology and Its Applications*. 51.

https://ecommons.udayton.edu/topology_conf/51

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