[Author] O. I. Reinov [Title] Approximation Properties: Topological Aspect (With Application to a Grothendieck Conjecture) [AMS Subj-class] 46B28 Spaces of operators; tensor products; approximation properties [Abstract] The natural generalizations of $\AP$ and $\BAP$ (the properties $\AP_p, BAP_p, AP_p^{dual}$ etc.) were considered earlier by P. Saphar, the author and others. All these properties were firstly defined in terms of some tensor products (generalizations of Grothendieck ones). One of the question of P. Saphar was to describe the $\AP_p$ in terms of something like "compact convergence" for absolutely $p$-summing operators. It was done earlier by the author of this paper. This paper is devoted, mainly, to the proceeding in the same direction for the class (ideal) $\Pi_p^d$ of dually absolutely-$p$-summing operators, and giving the connections between some notions of "compact convergence" of type $\pi_p^{dual}$ and the properties of tensor products, with applications. We will apply the results to the investigation of a natural question: when a Banach space has the generalized approximation properties (such as introduced by the author the properties $\AP^p)?$ We give here some sufficient and some necessary conditions for the space to have $\AP^p$ as well as construct some (counter)examples to the $\AP^p$-approximation problems. In the very end of the paper, we apply our results to give a new proof of a much more stronger result than one from my paper in C. R. Acad. Sc. Paris (concerning a conjecture of A. Grothendieck from his fundamental work on tensor products). Answering in negative to the Grothendieck question whether every weakly compact operator with the $\AP$ has also the $\BAP,$ we have constructed earlier the example of a compact operator with $\AP,$ but without $\BAP.$ Here, in Theorem 10, we show that there exist the operators of such a kind, belonging even to the classes of dually quasi-$p$-nuclear (hence, compact) operators. [Comments] AMSTeX, English, 18 pp. [Contact e-mail] orein51@mail.ru