Trivial fibrations with Tikhonov cube fiber (Q1083063)

Let \(\lambda\) be an arbitrary cardinal. We say that a map \(f: X\to Y\) satisfies the disjoint \(\lambda\)-approximation condition if for every collection \(\{\) \({\mathcal U}_{\alpha}:\alpha\in A\}\) of open covers of Y with \(| A| \leq \lambda\) there exist two maps \(g_ 1,g_ 2: X\to X\) such that: a) \(fg_ 1=f=fg_ 2\); b) \(g_ 1(X)\cap g_ 2(X)=\emptyset\); c) \(g_ i\) and \(id_ X\) are \({\mathcal U}_{\alpha}\)- close for each \(\alpha\in A\) \((I=1,2)\). The main result of the paper asserts that if \(\tau\) be an infinite cardinal then a soft map \(f: X\to Y\) between ANR-compacta is a trivial fibration with fiber \(I^{\tau}\) (the Tikhonov cube of weight \(\tau)\) iff all fibers of f have the weight \(\leq\tau\) and f satisfies the disjoint \(\lambda\)-approximation condition for each \(\lambda <\tau\). It should be observed that if \(\tau =\omega\) and X and Y are metrizable compacta then the above theorem coincides with the well-known result of Torunczyk-West concerning topological characterization of trivial fibration with Hilbert cube fiber.