A generalized extension or a g-extension of a space X is a pair (Y, f), where f : X → Y is a continuous mapping and the set f(X) is dense in Y . If f is an embedding, then Y is an extension of X and it is assumed that X = f(X) and f(x) = x for any x ∈ X. Let GE(X) be the set of all g-extension of the space X and E(X) be the set of all extensions of X. Obviously, E(X) ⊆ GE(X). If P is a topological property, then PGE(X) is the set of all g-extensions with the property P and PE(X) = E(X) ∩ PGE(X). A set L ⊆ PGE(X) is a complete lattice of g-extensions of X if for every non-empty set H ⊆ L we have (∨H)∩L ̸= ∅ and (∧H)∩L ̸= ∅. Let X be a T0-space. Let us denote by HGC(X) the set of the gcompactifications (bX, bX) of the space X for which bX is a Hausdorff space.Theorem 1. The set of HGC(X) is a complete lattice of g-extensions with the maximal element. The maximal element of the lattice HGC(X) is denoted by (βX, βX) and is called g-compactification Stone-˘Cech. Corollary 1. If f : X → Y is a continuous application, then there is a single continuous application βf : βX → βY for which βf ◦ βX = βY ◦ f. Corollary 2. If f : X → Y is a continuous application of the space X in the Hausdorff space and compact Y , then there is a single continuous application βf : βX → Y for which f = βf ◦ βX. Theorem 2. ([2], for T1-spaces). For any continuous application of space X in a Hausdorff and compact space Y , there is a single continuous application ωf : ωX → Y for which f = ωX|X.