Sewing in some classes of spaces (Q1358437)

There is a general rule in topology: if the intersection of two `nice' spaces \(X\) and \(Y\) is `nice', then the union \(X\cup Y\) is also `nice'. When `nice' means \(n\)-connected (locally \(n\)-connected), the intersection could be taken less `nicer' by one i.e. \((n-1)\)-connected (locally \((n-1)\)-connected). This can be generalized as follows. Let \(f:A\to Y\) be a map between `nice' spaces and let \(A\) be a closed subset of a `nice' space \(X\), then the sewing \(X\cup_fY\) is also `nice'. One of possible ways to see that consists of presentation of \(X\cup_fY\) as the natural cell-like image of the union \(X\cup M_f\) where \(M_f\) is the mapping cylinder. In this paper `nice' is considered to be either \(A(N)E\) or \(UV^n\) or \(n\)-movable. The sewing theorem for these cases is applied for proving a general theorem on preservation of these classes by covariant functors \(F: \text{Comp}\to \text{Comp}\) of a finite degree.