Abstract
In this paper a precompact shape theory is investigated. Necessary and sufficient conditions are found for which the precompact shapes of remainders are coinsided. An intrinsically characterization of Čech (co)homology groups of remainders is given. Border cohomological dimension, dim A ∞ X, and coefficient of border cyclicity, η A ∞ X, are defined and the inequality dim A ∞ X⩽dim A ( cX⧹ X) and the equality η A ∞ X= η A ( cX⧹ X) are proved for a space X normally adjoined to its remainder.
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