In this paper we define a space σ(X) for approximate system of compact spaces. The construction is due to H.Freudenthal for usual inverse sequence [4,pp. 153-156]. We establish the following properties of this space: (1) The space σ(X) is a paracompact space,(2) Moreover, if X is an approximate sequence of compact (metric) spaces,then σ(X)is a compact (metric) space (Lemma 2.4.). We give the following applications of the space σ(X): (3) If X is an approximate system of continua, then X=limX is a continuum (Theorem 3.1),(4) If X is an approximate system of hereditarily unicoherent spaces. then X=limX. is hereditarily unicoherent (Theorem 3.6.),(5) If X is an approximate system of the arboroids (generalized trees, trees, arcs) with monotone onto bonding mappings,then X=limX is an arboroid (generalized tree,tree,arc) (Theorems 3.18.,3.20.,3.21.,3.25.).