K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of the capacity of a compactum and asked some questions concerning properties of the capacity of compacta. In this paper, we give partial positive answers to three of these questions in some cases. In fact, by describing spaces homotopy dominated by Moore and Eilenberg-MacLane spaces, we obtain the capacity of a Moore space M(A, n) and an Eilenberg-MacLane space K(G, n). Also, we compute the capacity of the wedge sum of finitely many Moore spaces of different degrees and the capacity of the product of finitely many Eilenberg-MacLane spaces of different homotopy types. In particular, we give exact capacity of the wedge sum of finitely many spheres of the same or different dimensions.

K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of the capacity of a compactum and raised some interesting questions about it. In this paper, during computing the capacity of wedge sum of finitely many spheres of different dimensions and the complex projective plane, we give a negative answer to a question of Borsuk whether the capacity of a compactum determined by its homology properties.

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K. Borsuk in the seventies introduced the notions of capacity and depth of compacta together with some relevant problems. In this paper, first, we introduce the concepts of the (strong) capacity and the (strong) depth of an object in an arbitrary category. Then in the category of groups, we compute the (strong) capacity and the (strong) depth of some well-known groups. Finally, we find an upper bound for the depth of some classes of finite polyhedra which generalizes a result of D. Kolodziejczyk in this subject.The previous definition of the capacity of a compatum in the shape category of compacta coincides with Borsuk's definiton of the capacity (see [3]).2. The depth D(A) of an A ∈ Obj C is the least upper bound of the lengths of all chains for A. If this upper bound is infinite, we write D(A) = N 0 .3. A chain X k < s · · · < s X 1 d A, where X i ∈ Obj C for i = 1, · · · , k, is called an s-chain of length k for A ∈ Obj C. 4. The strong depth SD(A) of A is the least upper bound of the lengths of all s-chains for A. If this upper bound is infinite, we write SD(A) = N 0 .Note that our definition of strong depth of a compactum in the shape category of compacta coincides with Borsuk's definition of the depth of a compactum (for more details, see [3]).